Intersection of Line and Circle

What is the Intersection of a Line and Circle?

The intersection of a line and a circle refers to the common point or points that satisfy both the equation of the line and the equation of the circle. Depending on the position of the line relative to the circle, they may intersect at two points, touch at exactly one point, or have no common point at all. This concept is an important part of coordinate geometry and is frequently used in mathematics, engineering, computer graphics, and competitive examinations.

Intersection of a Line and Circle Meaning in Simple Words

In simple words, the intersection of a line and a circle means finding the point(s) where a straight line meets a circle. Sometimes the line cuts the circle, sometimes it just touches it, and sometimes it does not meet the circle at all.

Definition of Intersection of a Line and Circle

The intersection of a line and a circle is the set of points that satisfy both the equation of the line and the equation of the circle simultaneously.

If the equations are

$Ax+By+C=0$

and

$(x-h)^2+(y-k)^2=r^2$

then every common solution of these equations represents an intersection point.

Why Finding the Intersection is Important

Finding the intersection helps us:

• Determine whether a line is a secant, tangent, or external line.

• Solve coordinate geometry problems efficiently.

• Find points of contact between curves.

• Understand geometric relationships.

• Solve many JEE, NDA, CUET, and engineering entrance examination questions.

Real-Life Applications

The concept of line-circle intersection is used in:

• Road and railway design.

• Computer graphics.

• Robotics.

• GPS navigation.

• Engineering drawing.

• Architecture.

Basics of Coordinate Geometry

Before finding the points of intersection, it is important to understand the basic equations of lines and circles.

Equation of a Straight Line

A straight line can be written in several forms. The most common general form is

$Ax+By+C=0$

Other forms include:

• Slope-intercept form

$y=mx+c$

• Point-slope form

$y-y_1=m(x-x_1)$

Each form is useful depending on the given information.

Standard Equation of a Circle

A circle with centre

$(h,k)$

and radius

$r$

has the equation

$(x-h)^2+(y-k)^2=r^2$

If the centre is at the origin, the equation becomes

$x^2+y^2=r^2$

Centre and Radius of a Circle

Every circle is completely determined by:

• Its centre, which specifies the location.

• Its radius, which specifies the size.

These two quantities are essential while determining whether a line intersects the circle.

Position of a Point with Respect to a Circle

A point may lie:

• Inside the circle.

• On the circle.

• Outside the circle.

This depends on the distance of the point from the centre compared with the radius.

Different Types of Intersection

The number of common points depends on the relative position of the line and the circle.

Two Points of Intersection (Secant)

If the line passes through the circle, it intersects the circle at two distinct points.

Such a line is called a secant.

This happens when the perpendicular distance from the centre to the line is less than the radius.

One Point of Intersection (Tangent)

If the line touches the circle at exactly one point, it is called a tangent.

At the point of contact, the tangent is perpendicular to the radius.

No Point of Intersection

If the line lies completely outside the circle, it has no common point with the circle.

Such a line is called an external line.

Special Cases

Some important special situations include:

• The line passes through the centre of the circle.

• The line is parallel to the coordinate axes.

• The circle is centred at the origin.

• The line coincides with a diameter.

These cases often simplify calculations.

Methods to Find the Intersection of a Line and Circle

Several methods can be used depending on the form of the equations.

Algebraic Method

The algebraic method involves solving the equations of the line and the circle simultaneously.

After substituting one equation into the other, the resulting quadratic equation gives the intersection points.

This is the most commonly used method in competitive examinations.

Substitution Method

In this method, one variable from the line equation is expressed in terms of the other variable.

The resulting expression is substituted into the circle equation to obtain the required solution.

It is particularly useful when one equation is already solved for a variable.

Elimination Method

The elimination method removes one variable by combining the equations appropriately.

It is often preferred when both equations are given in general form.

Parametric Method

In advanced coordinate geometry, the line is represented using parametric equations.

Substituting the parametric coordinates into the circle equation helps determine the points of intersection efficiently.

This method is frequently used in higher mathematics and engineering.

Conditions for Intersection

The relative position of the line and circle can be determined without solving the equations completely.

Distance from the Centre to the Line

Discriminant Method

After substituting the line equation into the circle equation, a quadratic equation is obtained.

Its discriminant determines the number of intersection points.

• Positive discriminant: Two points.

• Zero discriminant: One point.

• Negative discriminant: No real point.

Tangency Condition

A line is tangent to the circle when

Distance from centre = Radius

or equivalently,

the quadratic equation obtained after substitution has discriminant equal to zero.

Secant Condition

A line is a secant when Distance from centre < Radius or the discriminant of the quadratic equation is positive.

This indicates two distinct intersection points.

Line and Circle

A circle is the locus of a moving point such that its distance from a fixed point is constant.

S is a circle with centre O and radius r, and L is a straight line in the plane of the circle.

Equation of circle $S: x^2+y^2=a^2 \quad \ldots$ (i)
Equation of line $L: y=m x+c$
To find their point(s) of intersection, we can solve these equations simultaneously

$\begin{aligned}
& x^2+(m x+c)^2=a^2 \\
& \left(1+m^2\right) x^2+2 m c x+c^2-a^2=0
\end{aligned}$

Case (1)

If line L intersects the circle S in two distinct points, then Equation (iii) will have two real and distinct roots

So, Discriminant of equation (iii) > 0

In this case, the line is a secant to the circle, i.e., it represents the equation of chord to the circle.

Case (2)

If the line L touches the circle, then Equation (iii) will have two equal real roots

So, Discriminant of equation (iii) = 0

$\begin{aligned} & \mathrm{B}^2-4 \mathrm{AC}=0 \\ & 4 \mathrm{~m}^2 \mathrm{c}^2-4\left(1+\mathrm{m}^2\right)\left(\mathrm{c}^2-\mathrm{a}^2\right)=0 \\ & \mathrm{a}^2=\frac{\mathrm{c}^2}{1+\mathrm{m}^2} \\ & \mathrm{c}^2=a^2\left(1+\mathrm{m}^2\right)\end{aligned}$

In this case, the line is tangent to the circle

This is also the condition of tangency to the circle.

Case (3)

If the line L and the circle S have no common points, then Equation (iii) will have imaginary roots

So, the Discriminant of equation (iii) < 0

Alternate method to check the position of the line with respect to the circle

We can find the distance OM between the line and the center of the circle, and compare it with the radius of the circle

• If OM < r, then the line is a chord (secant) to the circle
• If OM = r, then the line is tangent to the circle
• If OM > r, then the line does not intersect the circle at any point

Nature of the Intersection

The position of the line relative to the circle determines how they intersect. By examining the equation of the line and its distance from the centre of the circle, we can easily identify the nature of the intersection and simplify many coordinate geometry problems.

Line Passing Through the Centre

If the line passes through the centre of the circle, it always intersects the circle at two points.

The line divides the circle into two equal halves, and the chord formed is the diameter, which is the longest possible chord of the circle.

Such problems often become simpler because the midpoint of the two intersection points is the centre of the circle.

Line Parallel to an Axis

A line may be parallel to either the $x$-axis or the $y$-axis.

For example,

$x=a$ or $y=b$

These equations simplify substitution into the circle equation since only one variable needs to be determined.

The number of intersection points depends on the distance of the line from the centre of the circle.

Vertical and Horizontal Lines

Vertical lines have equations of the form $x=a$

while horizontal lines have equations of the form $y=b$

These special cases are easier to solve because they reduce the problem to solving a single-variable quadratic equation.

They frequently appear in coordinate geometry and competitive examinations.

Inclined Lines

An inclined line is any line that is neither vertical nor horizontal.

Its equation is generally written as $y=mx+c$ or $Ax+By+C=0$

To find the points of intersection, substitute the equation of the line into the circle equation and solve the resulting quadratic equation.

Properties of Line-Circle Intersection

Several important properties help determine the relationship between a line and a circle without performing lengthy calculations.

Number of Common Points

A line and a circle can have:

• Two common points, if the line is a secant.

• One common point, if the line is a tangent.

• No common point, if the line lies outside the circle.

Thus, the maximum possible number of intersection points is two.

Relationship Between Radius and Distance

The relationship between the radius and the perpendicular distance from the centre to the line determines the type of intersection.

• If Distance $<r$, the line is a secant.

• If Distance $=r$, the line is a tangent.

• If Distance $>r$, the line does not intersect the circle.

This property provides the quickest way to determine the nature of the intersection.

Geometrical Interpretation

Geometrically, the intersection points represent the locations where the straight line meets the circular boundary.

As the line moves closer to or farther from the centre:

• Two intersection points gradually approach each other.

• They coincide when the line becomes a tangent.

• They disappear once the line moves outside the circle.

This visual interpretation helps understand the transition between a secant, tangent, and external line.

Important Observations

Some useful observations include:

• Every tangent touches the circle at exactly one point.

• A secant always intersects the circle at two distinct points.

• A diameter is a special chord passing through the centre.

• Every diameter is formed by a secant passing through the centre.

• The tangent at any point on a circle is perpendicular to the radius drawn to the point of contact.

Applications of Intersection of a Line and Circle

The concept of line-circle intersection has applications far beyond coordinate geometry. It is widely used in mathematics, engineering, computer graphics, and navigation systems.

Intersection of a Line vs Tangent to a Circle

Although a tangent is a special type of line related to a circle, it differs significantly from a line that intersects the circle at two points.

Key Differences

An intersecting line (secant) cuts the circle at two distinct points.

A tangent touches the circle at exactly one point without entering its interior.

A secant forms a chord, whereas a tangent touches the circle only at the point of contact.

Similarities

Both:

• Are straight lines.

• Can be represented using linear equations.

• Interact with the circle.

• Are important concepts in coordinate geometry.

• Are frequently used in competitive examinations.

Formula Comparison

For a line

$Ax+By+C=0$

and a circle of radius $r$ with centre $(h,k)$,

Distance from the centre to the line is

$\frac{|Ah+Bk+C|}{\sqrt{A^2+B^2}}$

• If Distance $=r$ the line is a tangent.

• If Distance $<r$ the line is a secant.

• If Distance $>r$ there is no intersection.

Comparison Table

Best Books for Intersection of a Line and Circle

Understanding the intersection of a line and a circle requires a solid grasp of coordinate geometry, straight lines, and circles. The following books explain these concepts with numerous solved examples.

Shortcut Tips and Tricks

Many intersection problems become easier by first identifying the nature of the line with respect to the circle before solving the equations.

Important Formulae and Results

The following formulas are commonly used while solving problems involving the intersection of a line and a circle.

Solved Examples Based on the Intersection of a Line and a Circle

Example 1: The circle $x^2+y^2=4x+8y+5$ intersects the line $3x-4y=m$ at two distinct points if

1. $-85<m<-35$

2. $-35<m<15$

3. $15<m<65$

4. $35<m<85$

Solution:

The given circle is

$x^2+y^2-4x-8y-5=0$

The given line is

$3x-4y=m$

Substitute

$x=\frac{m+4y}{3}$

into the equation of the circle.

$\begin{aligned}
&\frac{(m+4y)^2}{9}+y^2-\frac{4(m+4y)}{3}-8y-5=0\
&\frac{25}{9}y^2+y\left(\frac{8m}{9}-\frac{40}{3}\right)+\left(\frac{m^2}{9}-5\right)=0
\end{aligned}$

Multiplying throughout by $9$,

$25y^2+8y(m-15)+(m^2-45)=0$

For the line to intersect the circle at two distinct points,

$D>0$

$\begin{aligned}
&[8(m-15)]^2-4(25)(m^2-45)>0\
&16m^2-3840m+23040-25m^2+1125>0
\end{aligned}$

On simplifying,

$(m+35)(m-15)<0$

Therefore,

$m\in(-35,15)$

Hence, the answer is option (2).

Example 2: Two common tangents to the circle $x^2+y^2=2a^2$ and the parabola $y^2=8ax$ are

1. $x=\pm(y+2a)$

2. $y=\pm(x+2a)$

3. $x=\pm(y+a)$

4. $y=\pm(x+a)$

Solution:

For the parabola

$y^2=4ax$

the tangent is

$y=mx+\frac{a}{m}$

Therefore, for

$y^2=8ax$,

the tangent is

$y=mx+\frac{2a}{m}$

For the circle

$x^2+y^2=2a^2$

a tangent is

$y=mx+a\sqrt{2(1+m^2)}$

Since the tangent is common,

$a\sqrt{2(1+m^2)}=\frac{2a}{m}$

Squaring,

$\begin{aligned}
&(m^2+1)m^2=2\
&m^4+m^2-2=0\
&(m^2+2)(m^2-1)=0
\end{aligned}$

Hence,

$m=\pm1$

Substituting,

$y=\pm(x+2a)$

Hence, the answer is option (2).

Example 3: If the chord $y=mx+1$ of the circle $x^2+y^2=1$ subtends an angle of measure $45^\circ$ at the major segment of the circle, then the value of $m$ is

Solution:

The given circle is

$x^2+y^2=1$

and the chord is

$y=mx+1$

Since the chord subtends an angle of $45^\circ$ at the major segment, it subtends an angle of $90^\circ$ at the centre.

Homogenizing the circle using

$y=mx+1$,

we get

$x^2+y^2=(y-mx)^2$

which simplifies to

$x^2(1-m^2)+2mxy=0$

For

$ax^2+by^2+2hxy=0$

to subtend an angle of $90^\circ$ at the origin,

$a+b=0$

Applying this condition,

$m=\pm1$

Hence, the value of $m$ is $\pm1$.

Example 4: If the line $3x-4y-\lambda=0$ touches the circle $x^2+y^2-4x-8y-5=0$ at $(a,b)$, then $\lambda+a+b$ is equal to:

Solution:

The centre of the circle is

$(2,4)$

and the radius is

$\sqrt{4+16+5}=5$

Since the line is tangent,

$\frac{|3(2)-4(4)-\lambda|}{\sqrt{3^2+(-4)^2}}=5$

$\Rightarrow\frac{|6-16-\lambda|}{5}=5$

Hence,

$\lambda=15$

or

$\lambda=-35$

The tangent at $(a,b)$ is

$\begin{aligned}
&xa+yb-2(x+a)-4(y+b)-5=0\
&\Rightarrow(a-2)x+(b-4)y-(2a+4b+5)=0
\end{aligned}$

Comparing with

$3x-4y-\lambda=0$,

$\frac{a-2}{3}=\frac{b-4}{-4}=\frac{2a+4b+5}{\lambda}=l$

Case I: $\lambda=15$

Then,

$a=3l+2,\qquad b=4-4l$

Substituting,

$2(3l+2)+4(4-4l)+5=15l$

$\Rightarrow l=1$

Thus,

$a=5,\qquad b=0$

Therefore,

$\lambda+a+b=15+5+0=20$

Case II: $\lambda=-35$

Similarly,

$25-10l=-35l$

$\Rightarrow l=-1$

Thus,

$a=-1,\qquad b=8$

Therefore,

$\lambda+a+b=-35-1+8=-28$

Hence, the required value is 20.

Example 5: The points of intersection of the line $ax+by=0,;(a\ne b)$ and the circle $x^2+y^2-2x=0$ are $A(\alpha,0)$ and $B(1,\beta)$. The image of the circle with $AB$ as a diameter in the line $x+y+2=0$ is

1. $x^2+y^2+3x+3y+4=0$

2. $x^2+y^2+3x+5y+8=0$

3. $x^2+y^2-5x-5y+12=0$

4. $x^2+y^2+5x+5y+12=0$

Solution:

The only possible values are

$\alpha=0,\qquad\beta=1$

The circle having $AB$ as diameter is

$\begin{aligned}
&(x-0)(x-1)+(y-0)(y-1)=0\
&x^2+y^2-x-y=0
\end{aligned}$

Reflecting this circle in the line

$x+y+2=0$

gives

$x^2+y^2+5x+5y+12=0$

Hence, the answer is option (4).

Related Topics to Intersection of a Line and Circle

The intersection of a line and a circle is closely related to equations of a circle, equations of a straight line, tangent to a circle, normal to a circle, chord of contact, coordinate geometry, and distance of a point from a line.