First slide

Analysis of two circles in circles

Question

The radical axis of the circles
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ and $2 (x^{2} + y^{2}) + 3 x + 2 y + 2 c = 0$
touches the circle $x^{2} + y^{2} + 2 x + 2 y + 1 = 0$ , if

Moderate

A

$g = \frac{4}{3}$ and $f = 2$
B

$g = \frac{4}{3}$ and $f = \frac{1}{2}$
C

$g = - \frac{3}{4}$ and $f = 2$
D

$g = \frac{3}{4}$ and $f = \frac{1}{2}$

Solution

The equation of radical axis is
$(2 g - \frac{3}{2}) x + (2 f - 1) y = 0$ (i)
It touches the circle $x^{2} + y^{2} + 2 x + 2 y + 1 = 0$
$\begin{array}{l} ∴ |\frac{- (2 g - \frac{3}{2}) - (2 f - 1)}{\sqrt{{(2 g - \frac{3}{2})}^{2} + (2 f - 1)^{2}}}| = 1 \\ \Rightarrow {(2 g + 2 f - \frac{5}{2})}^{2} = {(2 g - \frac{3}{2})}^{2} + (2 f - 1)^{2} \end{array}$
Clearly , $g = \frac{4}{3}$ and $f = \frac{1}{2}$

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