Line Equally Inclined with Two Lines

Edited By Komal Miglani | Updated on Jul 02, 2025 07:43 PM IST

In this article, we will cover the concept of Line Equally Inclined with two lines. This category falls under the broader category of Coordinate Geometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of ten questions have been asked on JEE MAINS( 2013 to 2023) from this topic.

This Story also Contains

What is Bisector?
Line Equally Inclined with two lines
Derivation of Line Equally Inclined with two lines
Solved Examples Based on Line Equally Inclined with two lines

Line Equally Inclined with Two Lines

What is Bisector?

The bisector is the locus of a point that moves in the plane of lines L₁ and L₂ such that lengths of perpendiculars drawn from it to the two given lines(L₁ and L₂) are equal.

Line Equally Inclined with two lines

A line equally inclined with two lines means the line which is the angle bisector of the angle made by two lines.

If the two lines with slope $m_1$ and $m_2$ are equally inclined to a line having slope $m$, then
$
\left(\frac{\mathbf{m}_1-\mathbf{m}}{1+\mathbf{m}_1 \mathbf{m}}\right)=-\left(\frac{\mathbf{m}_2-\mathbf{m}}{1+\mathbf{m}_2 \mathbf{m}}\right)
$

Derivation of Line Equally Inclined with two lines

Two lines with slopes $m_1$ and $m_2$ intersect at point A

As from the fig

$
\begin{aligned}
& \angle \mathrm{PAQ}=\angle \mathrm{QAR}=\angle \theta \\
& \tan (\angle \mathrm{PAQ})=\frac{\mathrm{m}_1-\mathrm{m}}{1+\mathrm{m}_1 \mathrm{~m}}=\tan \theta \\
& \text { and } \\
& \tan (\angle \mathrm{QAR})=\frac{\mathrm{m}-\mathrm{m}_2}{1+\mathrm{mm}_2}=\tan \theta
\end{aligned}
$

Hence,
$
\left(\frac{\mathbf{m}_1-\mathbf{m}}{1+\mathbf{m}_1 \mathbf{m}}\right)=-\left(\frac{\mathbf{m}_2-\mathbf{m}}{1+\mathbf{m}_2 \mathbf{m}}\right)
$

Solved Examples Based on Line Equally Inclined with two lines

Example 1: Equations of line $\mathrm{L}_1$ is $\sqrt{3} y-x+1=0$ and Equation of line $\mathrm{L}_2$ is $y-\sqrt{3} x+1=0$ then find the equation of Line which is equally inclined with both line and passing through the intersection point
Solution: If the two lines with slope $m_1$ and $m_2$ are equally inclined to a line having slope $m$, then
$
\left(\frac{\mathbf{m}_1-\mathbf{m}}{1+\mathbf{m}_1 \mathbf{m}}\right)=-\left(\frac{\mathbf{m}_2-\mathbf{m}}{1+\mathbf{m}_2 \mathbf{m}}\right)
$

Two lines with slopes $m_1$ and $m_2$ intersect at point $A$

As from the figure

Hence,
$
\begin{aligned}
& \left(\frac{\mathbf{m}_1-\mathbf{m}}{1+\mathbf{m}_1 \mathbf{m}}\right)=-\left(\frac{\mathbf{m}_2-\mathbf{m}}{1+\mathbf{m}_2 \mathbf{m}}\right) \\
& \quad \sqrt{3} y-x+1=0 \Rightarrow m_1=\frac{1}{\sqrt{3}} \\
& y-\sqrt{3} x+1=0 \Rightarrow m_2=\sqrt{3}^y y-\sqrt{3} x+1=0
\end{aligned}
$

If the two lines with slope $m_1$ and $m_2$ are equally inclined to a line having slope $m$, then $\left(\frac{m_1-m}{1+m_1 m}\right)=-\left(\frac{m_2-m}{1+m_2 m}\right)$
$
\begin{aligned}
& m= \pm 1 \\
& \quad\left(\frac{\frac{1}{\sqrt{3}}-\mathbf{m}}{1+\frac{1}{\sqrt{3}} \mathbf{m}}\right)=-\left(\frac{\sqrt{3}-\mathbf{m}}{1+\sqrt{3} \mathbf{m}}\right) \Rightarrow m^2=1 \\
& m
\end{aligned}
$

For Intersection point
$
\begin{aligned}
& \sqrt{3} y-x+1=0 \\
& y-\sqrt{3} x+1=0 \\
& x=\frac{\sqrt{3}-1}{2} \\
& y=\frac{3}{2}(1-\sqrt{3})
\end{aligned}
$

Equation of lines
$
\begin{aligned}
& y-\frac{3}{2}(1-\sqrt{3})=x-\frac{\sqrt{3}-1}{2} \\
& L_1: y=x+2-2 \sqrt{3} \\
& y-\frac{3}{2}(1-\sqrt{3})=-1\left(x-\frac{\sqrt{3}-1}{2}\right) \\
& L_2: y=-x+1-\sqrt{3}
\end{aligned}
$

Hence, the answer is $y=x+2-2 \sqrt{3}$

Example 2: A line is passing through (1,1) and $(2,1+\sqrt{3})$ and another line which is passing through (1,1) and $\left(2,1+\frac{1}{\sqrt{3}}\right)$ then find the equation of the line which is equally inclined with both lines.
Solution: If the two lines with slope $m_1$ and $m_2$ are equally inclined to a line having slope $m$, then
$
\left(\frac{\mathbf{m}_1-\mathbf{m}}{1+\mathbf{m}_1 \mathbf{m}}\right)=-\left(\frac{\mathbf{m}_2-\mathbf{m}}{1+\mathbf{m}_2 \mathbf{m}}\right)
$

Two lines with slopes $m_1$ and $m_2$ intersect at point $A$

As from the figure
$
\begin{aligned}
& \angle \mathrm{PAQ}=\angle \mathrm{QAR}=\angle \theta \\
& \tan (\angle \mathrm{PAQ})=\frac{\mathrm{m}_1-\mathrm{m}}{1+\mathrm{m}_1 \mathrm{~m}}=\tan \theta
\end{aligned}
$
and
$
\tan (\angle \mathrm{QAR})=\frac{\mathrm{m}-\mathrm{m}_2}{1+\mathrm{mm}_2}=\tan \theta
$

Hence,
$
\left(\frac{\mathbf{m}_1-\mathbf{m}}{1+\mathbf{m}_1 \mathbf{m}}\right)=-\left(\frac{\mathbf{m}_2-\mathbf{m}}{1+\mathbf{m}_2 \mathbf{m}}\right)
$

Slope of line $L_1=\frac{1+\sqrt{3}-1}{2-1}=\sqrt{3}=\tan 60^{\circ}$
Slope of line $L_2=\frac{1+\frac{1}{\sqrt{3}}-1}{2-1}=\frac{1}{\sqrt{3}}=\tan 30^{\circ}$
Slope of line which is equally inclined $=\tan 45^{\circ}=1$ and another line which is also equally inclined,slope is $=-1$ Equations of lines
$
\begin{aligned}
& L_1: y-1=x-1 \\
& \Rightarrow y-x=0 \\
& L_2: y-1=-1(x-1) \\
& \Rightarrow y+x-2=0
\end{aligned}
$

Hence, the answer is $y+x-2=0$

Example 3: Two equal sides of an isosceles triangle have the equations $7 x-y+3=0$ and $x+y=3$ and its third side passes through the point ( $-2,-1$ ). Then the equation of the third side.
Solution: If the two lines with slope $m_1$ and $m_2$ are equally inclined to a line having slope $m$, then
Two lines with slopes $m_1$ and $m_2$ intersect at point $A$

Let Equation of $A B$ is $7 x-y+3=0$ and equation of $A C x+y=3$
Slope of $\mathrm{AB}=7$ and Slope of $\mathrm{AC}=-1$
Let the equation of third side $B C=y=m x+c$
The angle between $A B$ and $B C=$ Angle between $A C$ and $B C$
$
\begin{aligned}
& \left|\frac{7-m}{1+7 m}\right|=\left|\frac{-1-m}{1-m}\right| \\
& -6 m^2-16 m+6=0 \\
& 3 m^2+8 m-3=0 \\
& m=-3 \text { and } m=1 / 3
\end{aligned}
$
equation of $B C$ is
$
(y-(-1))=-3(x-(-2)) \Rightarrow 3 x+y+7=0
$
or
$
(y-(-1))=\frac{1}{3}(x-(-2)) \Rightarrow x-3 y=1
$

Hence, the answer is $x-3 y=1$

Example 4: If $5 x^2+\lambda y^2=20$ represents a rectangular hyperbola, then $\lambda$ equal
Solution: The general equation of the second degree represents a rectangular hyperbola if $\Delta \neq 0, h^2>a b$ and coefficient of $x^2+$ coefficient of $y^2=0$.

Conditions	Nature of Conic
$\Delta \neq 0, h^2=a b$	Parabola
$\Delta \neq 0, h^2<a b$	Ellipse
$\Delta \neq 0, h^2>a b$	Hyperbola

The given equation represents a rectangular hyperbola if $\lambda+5=0$ i.e., $\lambda=-5$.
Hence, the answer is -5.

Example 5: A ray of light travelling along the line $2 \mathrm{x}-3 \mathrm{y}+5=0$ after striking a plane mirror lying along the line $\mathrm{x}+\mathrm{y}=2$ gets reflected. Find the equation of the straight line containing the reflected ray

Solution: The point of intersection of the lines $2 \mathrm{x}-3 \mathrm{y}+5=0$ and $\mathrm{x}+\mathrm{y}=2$ is $\left(\frac{1}{5}, \frac{9}{5}\right)$.
$\left(\frac{1}{5}, \frac{9}{5}\right)$ is the point of incidence.
The slope $m$ of the normal to the mirror (i.e. normal to the line $x+y=2$ ) is 1
Now the incident ray and reflected ray both are equally inclined to the normal and are on opposite sides of it.
The slope of the incident ray
$
\mathrm{m}_{\mathrm{I}}=\frac{2}{3}
$

Let the slope of the reflected ray be $=\mathrm{m}_2$
Then
$\frac{m_1-m}{1+m_1 \mathrm{~m}}=\frac{m-m_2}{1+m_2 m}$
$
\frac{\frac{2}{3}-1}{1+\frac{2}{3} \times 1}=\frac{1-m_2}{1+m_2 \times 1}
$

$\therefore \mathrm{m}_2=\frac{3}{2}, \quad \therefore$ the equation of the straight line containing the reflected ray is
$y-\frac{9}{5}=\frac{3}{2}\left(x-\frac{1}{5}\right)$
i.e. $\quad 3 x-2 y+3=0$

Hence, the answer is $3 x-2 y+3=0$

Frequently Asked Questions (FAQs)

1. What does it mean for a line to be equally inclined to two other lines?

A line is equally inclined to two other lines when it makes equal angles with both of them. This means that if you measure the angle between the equally inclined line and each of the other two lines, these angles will be the same.

2. Can a line be equally inclined to two parallel lines?

Yes, a line can be equally inclined to two parallel lines. In fact, any line that is perpendicular to the two parallel lines will be equally inclined to both of them, making 90° angles with each.

3. What is the relationship between the slopes of two lines if a third line is equally inclined to both?

If a line is equally inclined to two other lines, the product of its slope and the sum of the slopes of the other two lines equals the difference of their slopes. This relationship is expressed by the equation: m(m₁ + m₂) = 1 - m₁m₂, where m is the slope of the equally inclined line, and m₁ and m₂ are the slopes of the other two lines.

4. How can you determine if a line is equally inclined to two given lines without measuring angles?

You can determine if a line is equally inclined to two given lines by comparing the angles it makes with them using slope relationships. If the line satisfies the equation m(m₁ + m₂) = 1 - m₁m₂, where m is its slope and m₁ and m₂ are the slopes of the given lines, then it is equally inclined to both lines.

5. What is the geometric significance of a line being equally inclined to two other lines?

The geometric significance is that the equally inclined line creates a symmetry between the two other lines. It bisects the angle between the two lines if they intersect, or it forms congruent alternate angles if the two lines are parallel.

6. How does the concept of equally inclined lines relate to the angle bisector theorem?

The concept of equally inclined lines is closely related to the angle bisector theorem. When a line is equally inclined to two intersecting lines, it actually bisects the angle formed by those lines. This is because the equally inclined line makes equal angles with both intersecting lines, which is the definition of an angle bisector.

7. Can a horizontal line be equally inclined to two non-horizontal lines?

Yes, a horizontal line can be equally inclined to two non-horizontal lines, but only if those two lines are symmetrical about the vertical axis. In this case, the horizontal line would make equal angles (90°) with both non-horizontal lines.

8. Is it possible for a line to be equally inclined to more than two lines simultaneously?

Yes, it is possible for a line to be equally inclined to more than two lines simultaneously. This occurs when all the lines form equal angles with the equally inclined line. A common example is when multiple lines radiate from a single point, and another line intersects all of them at the same angle.

9. Can two lines be equally inclined to each other?

Two lines cannot be equally inclined to each other in the same way that a single line can be equally inclined to two others. The concept of equal inclination requires a third line to which the two lines are compared. However, two lines can have the same inclination with respect to a third line or to the horizontal.

10. How does the distance between two parallel lines affect a line equally inclined to both?

The distance between two parallel lines does not affect the angle of inclination of a line equally inclined to both. The equally inclined line will always be perpendicular to the parallel lines, regardless of the distance between them. However, the distance does affect where the equally inclined line intersects the parallel lines.

11. What is the difference between a line equally inclined to two lines and the perpendicular bisector of those lines?

A line equally inclined to two intersecting lines bisects the angle between them but doesn't necessarily bisect the distance between them. On the other hand, a perpendicular bisector is always perpendicular to the line segment joining two points and bisects this segment. They coincide only when the two original lines are perpendicular to each other.

12. What is the relationship between equally inclined lines and isosceles triangles?

An equally inclined line to two intersecting lines forms an isosceles triangle. The equally inclined line serves as the angle bisector of the vertex angle of this isosceles triangle, and the two equal angles it forms with the original lines correspond to the two equal base angles of the isosceles triangle.

13. How does the concept of equally inclined lines relate to the properties of kites?

In a kite, the diagonals are perpendicular to each other, and one diagonal bisects the other. This means that this bisecting diagonal is equally inclined to the two sides of the kite that it intersects. This property is crucial in proving various theorems related to kites and their symmetry.

14. What is the relationship between equally inclined lines and the angle addition formula in trigonometry?

The concept of equally inclined lines is related to the angle addition formula in trigonometry. When a line is equally inclined to two others, it bisects the angle between them. This bisection property is fundamental to deriving the angle addition formulas for sine and cosine, which are crucial in trigonometry.

15. How is the angle of inclination related to the slope of a line?

The angle of inclination of a line is directly related to its slope. The tangent of the angle of inclination equals the absolute value of the slope. For example, if a line has a slope of 1, its angle of inclination is 45°, because tan(45°) = 1.

16. What happens to the equally inclined line as the angle between the two original lines approaches 0°?

As the angle between the two original lines approaches 0°, meaning the lines are getting closer to being parallel, the equally inclined line approaches the perpendicular to these lines. In the limit, when the lines are parallel, the equally inclined line is exactly perpendicular to both.

17. Can the concept of equally inclined lines be extended to three-dimensional geometry?

Yes, the concept of equally inclined lines can be extended to three-dimensional geometry. In 3D space, a line can be equally inclined to two or more planes. For example, the axis of a right circular cone is equally inclined to all lines on the surface of the cone. This extension is important in studying 3D shapes, crystallography, and spatial geometry.

18. Can the principle of equally inclined lines be applied to non-Euclidean geometries?

Yes, the principle of equally inclined lines can be extended to non-Euclidean geometries, but with some modifications. In spherical geometry, for instance, the concept applies to great circles on a sphere. In hyperbolic geometry, the notion of "equally inclined" needs to be redefined in terms of the geometry's unique properties. These extensions are important in advanced geometric studies and in some areas of physics.

19. How does the concept of equally inclined lines relate to the properties of medians in a triangle?

While medians in a triangle are not necessarily equally inclined to the sides they connect, they do have a related property. The centroid of a triangle, where the three medians intersect, divides each median in a 2:1 ratio. This point has the unique property that any line through it will divide the area of the triangle in a constant ratio, which is related to the concept of equal inclination.

20. How can the concept of equally inclined lines be used in computer graphics and animation?

In computer graphics and animation, the concept of equally inclined lines is used in various ways. It's crucial in creating symmetrical designs, in implementing reflection and rotation algorithms, and in calculating light reflections for realistic rendering. It's also used in path-finding algorithms and in creating smooth transitions between different orientations of objects.

21. How does the concept of equally inclined lines apply to the study of parabolas?

In a parabola, any tangent line is equally inclined to the focal axis and the line joining the point of tangency to the focus. This property is fundamental to the reflective properties of parabolas and is used in designing parabolic antennas, solar collectors, and headlights. It's also key to understanding the path of projectiles in physics.

22. How does the slope of an equally inclined line relate to the slopes of the two lines it's inclined to?

The slope of an equally inclined line (m) is related to the slopes of the two lines it's inclined to (m₁ and m₂) by the equation: m = (m₁ - m₂) / (1 + m₁m₂). This formula allows you to calculate the slope of the equally inclined line if you know the slopes of the other two lines.

23. How can the concept of equally inclined lines be applied in real-world scenarios?

The concept of equally inclined lines has various real-world applications. In architecture, it's used for designing symmetrical structures. In optics, it's crucial for understanding light reflection. In engineering, it's applied in designing ramps, roofs, and other inclined structures. In navigation, it's used for triangulation and determining positions.

24. Can a vertical line be equally inclined to two non-vertical lines?

Yes, a vertical line can be equally inclined to two non-vertical lines, but only if those two lines are symmetrical about the horizontal axis. In this case, the vertical line would make equal angles (90°) with both non-vertical lines.

25. How does the concept of equally inclined lines apply in coordinate geometry?

In coordinate geometry, equally inclined lines can be analyzed using their equations and slopes. The condition for a line to be equally inclined to two others can be expressed algebraically using the slopes of the lines. This allows for precise calculations and proofs involving equally inclined lines in the coordinate plane.

26. How can you construct a line equally inclined to two given lines using only a compass and straightedge?

To construct a line equally inclined to two given intersecting lines using compass and straightedge: 1) Draw an arc centered at the intersection point, cutting both lines. 2) With the same radius, draw arcs centered where the first arc intersects the lines. 3) These new arcs intersect at two points. 4) The line through these points and the original intersection point is equally inclined to the given lines.

27. What is the significance of equally inclined lines in the study of conics?

In the study of conics, equally inclined lines play a role in understanding properties of parabolas, ellipses, and hyperbolas. For instance, in a parabola, any tangent line is equally inclined to the focal axis and the line joining the point of tangency to the focus. This property is key to understanding reflection properties of parabolic mirrors and antennas.

28. How does the concept of equally inclined lines relate to the properties of rhombuses?

In a rhombus, the diagonals bisect each other at right angles. This means that each diagonal is equally inclined to the two sides it intersects. This property is fundamental to many proofs involving rhombuses and helps explain their symmetry properties.

29. Can a curve be equally inclined to two straight lines?

While the concept of equal inclination is typically applied to straight lines, it can be extended to curves in certain contexts. For example, a parabola can be considered equally inclined to two lines if its axis of symmetry bisects the angle between those lines. However, the interpretation may vary depending on the specific curve and context.

30. How does the idea of equally inclined lines relate to the concept of symmetry in geometry?

Equally inclined lines are closely related to symmetry in geometry. A line equally inclined to two others often acts as a line of symmetry, especially when the two original lines intersect. This concept is crucial in understanding and proving properties of various symmetrical shapes and in analyzing symmetry in more complex geometric figures.

31. What is the relationship between equally inclined lines and the properties of isosceles trapezoids?

In an isosceles trapezoid, the non-parallel sides are equally inclined to both parallel sides. This property is key to understanding the symmetry of isosceles trapezoids and is used in proving various theorems about their angles and diagonals. The line of symmetry in an isosceles trapezoid is equally inclined to both non-parallel sides.

32. How can the concept of equally inclined lines be used to solve problems involving angle bisectors?

The concept of equally inclined lines is directly applicable to problems involving angle bisectors. Since an angle bisector is equally inclined to the sides of the angle it bisects, properties of equally inclined lines can be used to solve problems about angle bisectors, such as finding their length, proving their uniqueness, or determining their intersection points with other lines.

33. What is the connection between equally inclined lines and the reflection of light?

The principle of equally inclined lines is fundamental in optics, particularly in the study of light reflection. When light reflects off a surface, the angle of incidence equals the angle of reflection. This means that the incident ray and the reflected ray are equally inclined to the normal (perpendicular) line at the point of reflection. This principle is crucial in designing mirrors, lenses, and other optical devices.

34. How does the concept of equally inclined lines apply to the study of parallel lines cut by a transversal?

When a transversal cuts two parallel lines, it creates several pairs of equal angles. The alternate interior angles, alternate exterior angles, and corresponding angles are all equal. This means that the transversal is equally inclined to both parallel lines. Understanding this helps in proving theorems about parallel lines and in solving related geometric problems.

35. How does the slope of an equally inclined line change as the angle between the two original lines varies?

As the angle between the two original lines increases, the slope of the equally inclined line generally decreases in absolute value. When the original lines are nearly parallel (small angle), the equally inclined line is nearly perpendicular to them (high absolute slope). As the angle approaches 180°, the equally inclined line becomes more horizontal (slope approaching zero).

36. What is the relationship between equally inclined lines and the properties of regular polygons?

In regular polygons, lines drawn from the center to the vertices are equally inclined to adjacent sides. This property is key to understanding the symmetry of regular polygons. Additionally, the angle bisectors of a regular polygon are equally inclined to the sides they intersect, which is crucial in proving various theorems about regular polygons.

37. How can the concept of equally inclined lines be used in solving optimization problems?

The concept of equally inclined lines can be used in optimization problems, particularly those involving minimizing or maximizing distances or angles. For example, finding the shortest path that touches two intersecting lines often involves a path that is equally inclined to both lines. This principle is applied in fields like operations research and computer graphics.

38. What is the significance of equally inclined lines in the study of tangent lines to circles?

In circle geometry, a tangent line at any point is perpendicular to the radius drawn to that point. This means that the tangent line is equally inclined to any two diameters of the circle that are symmetrical about this radius. This property is fundamental in proving theorems about tangents and in solving problems related to circles and their tangents.

39. How does the concept of equally inclined lines relate to the properties of cyclic quadrilaterals?

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), opposite angles are supplementary. This means that any line drawn from a vertex to the opposite side will be equally inclined to the two adjacent sides. This property is crucial in proving various theorems about cyclic quadrilaterals and in solving related geometric problems.

40. What is the relationship between equally inclined lines and the concept of orthocenter in a triangle?

The orthocenter of a triangle, which is the point where the three altitudes intersect, has an interesting relationship with equally inclined lines. Each altitude is perpendicular to its corresponding side, making it equally inclined to the other two sides. Understanding this helps in proving properties of the orthocenter and in solving advanced triangle problems.

41. What is the significance of equally inclined lines in the study of ellipses?

In an ellipse, the normal line at any point is equally inclined to the two lines drawn from that point to the foci. This property, known as the optical property of ellipses, is crucial in understanding how ellipses reflect light and sound. It's used in designing elliptical reflectors and in studying planetary orbits, which are elliptical.

42. What is the relationship between equally inclined lines and the properties of similar triangles?

In similar triangles, corresponding angles are equal, which means that corresponding sides are equally inclined to each other. This property is fundamental in proving the similarity of triangles and in solving problems involving proportional segments. Understanding this relationship helps in applying the concept of equally inclined lines to more complex geometric figures.

43. How can the concept of equally inclined lines be applied in surveying and mapmaking?

In surveying and mapmaking, the concept of equally inclined lines is used in triangulation methods. By measuring angles from known points to an unknown point, surv