Tangents and Normals to a Circle: JEE Mathematics Mastery

Welcome to this module on Tangents and Normals to a Circle, a crucial topic for JEE Mathematics. Understanding these concepts is key to solving a wide range of problems involving circles and their interactions with lines. We'll explore their definitions, properties, and methods for finding their equations.

Understanding Tangents

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of tangency. A fundamental property is that the radius drawn to the point of tangency is perpendicular to the tangent line.

The tangent is a line that intersects the circle at precisely one point.

Imagine a line just grazing the edge of a circle. That's a tangent! It has only one point in common with the circle.

Mathematically, a tangent line to a circle is a line that intersects the circle at exactly one point. If a line intersects a circle at two points, it is called a secant. The point where the tangent touches the circle is known as the point of contact or point of tangency.

What is the defining characteristic of a tangent line to a circle?

A tangent line touches the circle at exactly one point.

Finding the Equation of a Tangent

There are several ways to find the equation of a tangent to a circle, depending on the information given. We'll cover tangents at a point on the circle, tangents with a given slope, and tangents from an external point.

Tangent at a Point on the Circle

For a circle with equation $x^2 + y^2 = r^2$ , the equation of the tangent at a point $(x_1, y_1)$ on the circle is $xx_1 + yy_1 = r^2$ . If the circle's equation is $(x-h)^2 + (y-k)^2 = r^2$ , the tangent at $(x_1, y_1)$ is $(x-h)(x_1-h) + (y-k)(y_1-k) = r^2$ .

Tangent with a Given Slope

For the circle $x^2 + y^2 = r^2$ , the equations of the tangents with slope $m$ are $y = mx \pm r\sqrt{1+m^2}$ . For $(x-h)^2 + (y-k)^2 = r^2$ , the tangents are $y-k = m(x-h) \pm r\sqrt{1+m^2}$ .

Tangent from an External Point

When a tangent is drawn from an external point $(x_1, y_1)$ to a circle, there are generally two tangents. One method to find their equations is to assume the equation of the tangent with an unknown slope and use the condition that the distance from the center of the circle to the tangent is equal to the radius.

Consider a circle centered at the origin with radius 'r'. A tangent line at point P(x1, y1) on the circle is perpendicular to the radius OP. The slope of OP is y1/x1. The slope of the tangent is therefore -x1/y1. Using the point-slope form of a line, the tangent equation is y - y1 = (-x1/y1)(x - x1), which simplifies to yy1 - y1^2 = -xx1 + x1^2. Rearranging gives xx1 + yy1 = x1^2 + y1^2. Since (x1, y1) is on the circle, x1^2 + y1^2 = r^2. Thus, the tangent equation is xx1 + yy1 = r^2.

📚

Text-based content

Library pages focus on text content

Understanding Normals

A normal to a circle is a straight line that is perpendicular to the tangent at the point of tangency. A key property of normals is that they always pass through the center of the circle.

The normal to a circle at any point is the line passing through that point and the center of the circle.

Think of the normal as a line that goes straight through the point of contact and heads directly towards the center of the circle.

The normal line at a point on a circle is defined as the line perpendicular to the tangent at that point. Crucially, for any circle, all normal lines pass through the center of the circle. This simplifies finding the equation of a normal significantly.

What is the defining characteristic of a normal line to a circle?

A normal line is perpendicular to the tangent at the point of tangency and passes through the center of the circle.

Finding the Equation of a Normal

If the circle's equation is $(x-h)^2 + (y-k)^2 = r^2$ , its center is $(h, k)$ . The equation of the normal at any point $(x_1, y_1)$ on the circle is simply the equation of the line passing through $(x_1, y_1)$ and $(h, k)$ . This can be found using the two-point form of a line: $\frac{y - k}{x - h} = \frac{y_1 - k}{x_1 - h}$ .

Feature	Tangent	Normal
Definition	Line touching circle at one point	Line perpendicular to tangent at point of contact
Relation to Center	Perpendicular to radius at point of tangency	Always passes through the center
Number of points common with circle	Exactly one	Infinite (if it's a diameter), but typically considered in relation to the point of tangency

Remember: The normal is the line that connects the point of tangency directly to the center of the circle.

Key Takeaways and Practice

Mastering tangents and normals involves understanding their geometric properties and applying the correct algebraic formulas. Practice problems involving finding tangent equations at a point, with a given slope, and from an external point. For normals, focus on identifying the center and using the two-point form of a line.

Learning Resources

Tangents and Normals to a Circle - Vedantu(documentation)

Provides a comprehensive overview of tangents and normals to a circle, including formulas and examples relevant to competitive exams.

NCERT Mathematics Class 11 - Chapter 10: Straight Lines(documentation)

This official NCERT textbook chapter covers coordinate geometry, including tangents and normals to circles, offering foundational knowledge.

Tangents to Circles - Brilliant.org(documentation)

Explains the concept of tangents to circles with interactive examples and clear explanations suitable for competitive exam preparation.

Coordinate Geometry: Tangents and Normals - Byju's(blog)

A detailed explanation of tangents and normals with solved examples, focusing on the application of formulas for JEE preparation.

JEE Mathematics: Circle - Tangents and Normals(documentation)

Focuses on the equation of the tangent to a circle at a given point, a common problem type in JEE.

Understanding Normals to a Circle - Toppr(documentation)

Explains the concept and equation of a normal to a circle, highlighting its relationship with the center.

Coordinate Geometry - Circles | JEE Main(video)

A video tutorial covering circles in coordinate geometry for JEE Main, likely including tangents and normals.

JEE Advanced Mathematics - Circle Properties(wikipedia)

While not a direct tutorial, StackExchange provides in-depth discussions and proofs related to circle properties, useful for advanced understanding.

Practice Problems: Tangents and Normals to Circles(blog)

A collection of questions and discussions on tangents and normals, offering diverse problem-solving scenarios.

Coordinate Geometry Formulas for Circles - JEE(blog)

A summary of essential formulas for circles in coordinate geometry, including those for tangents and normals, crucial for quick recall.