Intersections of a Line and a Circle

Understanding how a straight line interacts with a circle is a fundamental concept in coordinate geometry, crucial for solving problems in competitive exams like JEE. This module explores the different possibilities of intersection and the algebraic methods to determine them.

Geometric Possibilities

Geometrically, a line can interact with a circle in three distinct ways:

Scenario	Number of Intersection Points	Geometric Interpretation
Secant Line	Two	The line passes through the interior of the circle, crossing its boundary at two distinct points.
Tangent Line	One	The line touches the circle at exactly one point, known as the point of tangency. The line does not enter the circle's interior.
Exterior Line	Zero	The line does not touch the circle at all; it lies entirely outside the circle's boundary.

Algebraic Approach to Finding Intersections

To find the exact points of intersection algebraically, we use a system of equations. We have the equation of the circle and the equation of the line. By substituting the expression for one variable from the linear equation into the circular equation, we obtain a quadratic equation in a single variable. The nature of the roots of this quadratic equation tells us about the number of intersection points.

The discriminant of the resulting quadratic equation determines the number of intersection points.

When you substitute a line into a circle's equation, you get a quadratic. The discriminant (Δ = b² - 4ac) of this quadratic reveals the intersection type: Δ > 0 means two points (secant), Δ = 0 means one point (tangent), and Δ < 0 means no points (exterior).

Let the equation of the circle be $(x-h)^2 + (y-k)^2 = r^2$ and the equation of the line be $y = mx + c$ . Substituting $y$ from the line equation into the circle equation gives: $(x-h)^2 + (mx+c-k)^2 = r^2$ . Expanding and rearranging this will result in a quadratic equation of the form $Ax^2 + Bx + C = 0$ . The discriminant, $\Delta = B^2 - 4AC$ , dictates the number of real solutions for $x$ , which correspond to the x-coordinates of the intersection points.

If $\Delta > 0$ , there are two distinct real roots for $x$ , meaning the line intersects the circle at two points.
If $\Delta = 0$ , there is exactly one real root for $x$ , meaning the line is tangent to the circle at one point.
If $\Delta < 0$ , there are no real roots for $x$ , meaning the line does not intersect the circle.

What does a discriminant of zero (Δ = 0) signify when finding the intersection of a line and a circle?

It signifies that the line is tangent to the circle, intersecting it at exactly one point.

Finding the Point(s) of Intersection

Once the quadratic equation is solved for $x$ (if $\Delta \ge 0$ ), the corresponding $y$ values can be found by substituting these $x$ values back into the equation of the line ( $y = mx + c$ ). This gives the coordinates of the intersection point(s).

A shortcut for tangency: If a line $y = mx + c$ is tangent to a circle $(x-h)^2 + (y-k)^2 = r^2$ , the distance from the center of the circle $(h, k)$ to the line is equal to the radius $r$ . The distance formula from a point $(x_0, y_0)$ to a line $Ax + By + C = 0$ is $\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$ .

Consider a circle centered at the origin $(0,0)$ with radius $r$ , given by $x^2 + y^2 = r^2$ . Let a line be $y = mx + c$ . Substituting $y$ into the circle equation gives $x^2 + (mx+c)^2 = r^2$ . This expands to $x^2 + m^2x^2 + 2mcx + c^2 = r^2$ , which rearranges to $(1+m^2)x^2 + (2mc)x + (c^2 - r^2) = 0$ . This is a quadratic equation in $x$ . The discriminant is $\Delta = (2mc)^2 - 4(1+m^2)(c^2 - r^2)$ . For tangency, $\Delta = 0$ . This leads to the condition for tangency: $r^2(1+m^2) = c^2$ . This visual shows the geometric interpretation of the discriminant: a positive discriminant means the line cuts through the circle, a zero discriminant means it just touches, and a negative discriminant means it misses entirely.

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Example Problem

Find the points of intersection of the line $y = x + 1$ and the circle $x^2 + y^2 = 5$ .

Substitute $y = x + 1$ into the circle equation: $x^2 + (x+1)^2 = 5$ $x^2 + (x^2 + 2x + 1) = 5$ $2x^2 + 2x + 1 = 5$ $2x^2 + 2x - 4 = 0$ $x^2 + x - 2 = 0$ Factoring the quadratic: $(x+2)(x-1) = 0$ . So, $x = -2$ or $x = 1$ .

If $x = -2$ , then $y = -2 + 1 = -1$ . Point: $(-2, -1)$ . If $x = 1$ , then $y = 1 + 1 = 2$ . Point: $(1, 2)$ .

The points of intersection are $(-2, -1)$ and $(1, 2)$ .

Learning Resources

Intersection of a Line and a Circle - Maths is Fun(documentation)

Provides a clear explanation of the geometric and algebraic methods for finding line-circle intersections, with examples.

Coordinate Geometry: Intersection of a Line and a Circle - Byju's(blog)

Explains the concept with a focus on the discriminant and provides solved examples relevant to competitive exams.

JEE Mathematics: Straight Lines and Circles - Vedantu(documentation)

A comprehensive overview of coordinate geometry for JEE, including sections on lines and circles and their interactions.

Circle and Line Intersection - Wolfram MathWorld(documentation)

A more advanced and detailed mathematical treatment of the intersection problem, including formulas and derivations.

Coordinate Geometry - Circles - Khan Academy(video)

A video tutorial explaining how to find intersection points of circles and lines using substitution.

Tangency Condition for a Line to a Circle - Toppr(blog)

Focuses specifically on the condition for a line to be tangent to a circle, a key aspect of intersection problems.

Coordinate Geometry - Straight Lines - Tutorialspoint(documentation)

Covers the basics of straight lines, which is essential for understanding the line equation part of the intersection problem.

JEE Main 2024 Mathematics Syllabus - Circles(documentation)

Official syllabus for JEE Main, confirming the importance of circles and their properties in the exam.

Geometric Interpretation of Discriminant - Brilliant.org(documentation)

Explains the geometric meaning of the discriminant in quadratic equations, directly applicable to line-circle intersections.

Solving Systems of Equations - Purplemath(documentation)

General guidance on solving systems of equations, including substitution methods, which are fundamental to this topic.