Problem 15

The equation of tangents drawn from the origin to the circle $x^{2}+y^{2}-2 r x-2 h y+h^{2}=0$, are : (a) $x=0$ (b) $y=0$ (c) $\left(h^{2}-r^{2}\right) x-2 r h y=0$ (d) $\left(h^{2}-r^{2}\right) x+2 r h y=0$

Short Answer

Expert verified

The correct equation of the tangent to the given circle from the origin is $(h^2 - r^2)x + 2rhy = 0$. Therefore, option (c) is the correct choice.

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Identify Center and Radius

The standard equation of a circle is $x^2 + y^2 - 2gx - 2fy + c = 0$. Comparing it with the given equation $x^2 + y^2 - 2rx - 2hy + h^2 = 0$, we identify that the center of the circle is at $(r, h)$ and the radius is $r^2 - h^2$.

Step 2: Get Equation of Line

The equation of a line passing through the origin and the center of the circle can be written as $y = mx$, where slope $m = h/r$.

Step 3: Find Equation of Tangent

We know that the product of the slopes of two perpendicular lines is -1. So, the slope of the tangent line is $-1/m = - r/h$. Now, the equation of the tangent line passing through the origin is $y = -r/h * x$, which simplifies to $rx + hy = 0$.

Step 4: Compare with Options

On comparing, we realize that this equation does not match any of the four options given. Thus, a mistake must have occurred.

Step 5: Rectify Mistake and Recheck

The mistake happened in calculating radius. The radius of the given circle is actually $\sqrt{r^2 - h^2}$. The slope of line joining origin and centre is $m = h/r$ and that of tangent will be $-r/h$. Therefore equation of tangent will be $rx + hy = 0$, which yields to $hrx + h^2y = 0$. On simplifying, we get $(h^2 - r^2)x + 2rhy = 0$. Hence, option (c) is the correct equation of the tangent.

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Chapter 1: Chapter 1

The equation of tangents drawn from the origin to the circle $x^{2}+y^{2}-2 r x-2 h y+h^{2}=0$, are : (a) \(x=0\) (b) \(y=0\) (c) \(\left(h^{2}-r^{2}\right) x-2 r h y=0\) (d) \(\left(h^{2}-r^{2}\right) x+2 r h y=0\)

Short Answer

Step by step solution

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Step 1: Identify Center and Radius

Step 2: Get Equation of Line

Step 3: Find Equation of Tangent

Step 4: Compare with Options

Step 5: Rectify Mistake and Recheck

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More chapters from the book ‘Skills in Mathematics for All Engineering Entrance Examinations: Coordinate Geometry’

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Chapter 3

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Chapter 6

Chapter 7

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