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Common Mistakes in CAT Reading Comprehension Questions; Avoid These RC Errors for CAT 2026
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CAT Equation of Tangent - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Quick Facts

  • 5 Questions around this concept.

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QuestionEASY

Find the equation to the tangent of circle x^2+y^2=25 at the point (4, 3).

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Equation of tangent to the circle x^2+y^2+4x+4y+16=0 at point (x_1,y_1) is:

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Concepts Covered - 1

Equation of Tangent

In the previous concept we studied that the tangent at any point of a circle is perpendicular to the radius through the point of contact. We will use this concept to find the equation of a tangent to a circle.

Let the circle with center C(-g, -f) and a tangent PT with Point P (x1, y1) lies on the circle. From the theorem, PT is perpendicular to CP.

\text { Slope of } \mathrm{CP}=\frac{\mathrm{y}_{1}-(-\mathrm{f})}{\mathrm{x}_{1}-(-\mathrm{g})}=\frac{\mathrm{y}_{1}+\mathrm{f}}{\mathrm{x}_{1}+\mathrm{g}}

Also we know that if two lines are perpendicular, then multiplication of their slopes is equal to -1.

i.e.

\\\text{(slope of PT) x (slope of CP) = -1} \\\Rightarrow \text{(slope of PT)}\times\left ( \frac{\mathrm{y}_{1}+\mathrm{f}}{\mathrm{x}_{1}+\mathrm{g}} \right )=-1\\\Rightarrow \text{(slope of PT)}=-\left(\frac{\mathrm{x}_{1}+\mathrm{g}}{\mathrm{y}_{1}+\mathrm{f}}\right)

Now, the equation of the tangent at point P(x1, y1) is

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\left(\mathrm{y}-\mathrm{y}_{1}\right)=-\left(\frac{\mathrm{x}_{1}+\mathrm{g}}{\mathrm{y}_{1}+\mathrm{f}}\right)\left(\mathrm{x}-\mathrm{x}_{1}\right) \\ \Rightarrow \quad\left(\mathrm{y}-\mathrm{y}_{1}\right)\left(\mathrm{y}_{1}+\mathrm{f}\right)+\left(\mathrm{x}_{1}+\mathrm{g}\right)\left(\mathrm{x}-\mathrm{x}_{1}\right)=0 \\ \Rightarrow \quad \mathrm{xx}_{1}+\mathrm{yy}_{1}+\mathrm{gx}+\mathrm{fy}=\mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}+\mathrm{gx}_{1}+\mathrm{fy}_{1}

Let’s simplify the above equation

add \mathrm{g x_{1}+f y_{1}+c} both side, we get

\\\Rightarrow \quad \mathrm{x x_{1}+y y_{1}+g\left(x+x_{1}\right)+f\left(y+y_{1}\right)+c=x_{1}^{2}+y_{1}^{2}+2 g x_{1}+2 f y_{1}+c}\\\Rightarrow\mathrm{\;\;\;}\mathrm{ x x_{1}+y y_{1}+g\left(x+x_{1}\right)+f\left(y_{1}+ y\right)+c=0 }

As we know that the general equation of circle x2+y2+2gx+2fy+c=0 and the point (x1,y1) lies on the circumference of the circle.

Also \mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}+2 \mathrm{gx}_{1}+2 \mathrm{fy}_{1}+\mathrm{c}=0  as point (x1,y1) lies on the circle

Hence, \mathrm{x x_{1}+y y_{1}+g\left(x+x_{1}\right)+f\left(y_{1}+ y\right)+c=0} is the equation of tangent at point (x1,y1) of circle having centre at (-g, -f). 

The equation of the tangent to the circle \mathrm{x^2+y^2=a^2} at point (x1,y1) is \mathrm{xx_1+yy_1=a^2}.

Study other Related Concepts

CAT Equation of Tangent Current Topic

CAT Basic Terms and Definitions of Lines and Angles
CAT Various Types of Angle
CAT Linear Pair of Angles
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CAT Corresponding Angle Axioms
CAT Alternate Interior Angle Theorem
CAT Interior Angle Theorem
CAT Parallel and a Transversal Lines
CAT Introduction to Triangles
CAT Congruence of Triangles

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