First slide

Definition of a circle

Question

Circle $(s)$ touching $x -$ axis at a distance $3$ from the origin and having an intercept of length $2 \sqrt{7}$ on $y -$ axis, is (are)

Moderate

A

$x^{2} + y^{2} - 6 x \pm 8 y + 9 = 0$
B

$x^{2} + y^{2} - 6 x \pm 7 y + 9 = 0$
C

$x^{2} + y^{2} + 6 x \pm 8 y + 9 = 0$
D

$x^{2} + y^{2} - 8 x \pm 6 y + 9 = 0$

Solution

Let the equation of the circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0 .$ It touches $x -$ axis at a distance $3$ from the origin. Therefore, $c = g^{2}$ and the circle passes through $(\pm 3, 0) .$
$\begin{array}{ll} ∴ 9 \pm 6 g + c = 0 \\ \Rightarrow 9 \pm 6 g + g^{2} = 0 \Rightarrow (g \pm 3)^{2} = 0 \Rightarrow g = - 3 \\ ∴ c = g^{2} \Rightarrow c = 9 \end{array}$
The circle cuts an intercept of length $2 \sqrt{7}$ on $y -$ axis.
$∴ 2 \sqrt{f^{2} - c} = 2 \sqrt{7} \Rightarrow f^{2} - 9 = 7 \Rightarrow f = \pm 4$
Hence, the equations of the circles are
$x^{2} + y^{2} - 6 x \pm 8 y + 9 = 0$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App