Question

Find the equation of the circle which passes through the point (1, 1) & which touches the circle $x^2+y^2+4x-6y-3=0$ at the point $(2,3)$ on it.

• A. $x^2+y^2+x-6y+3=0$
• B. $x^2+y^2+x-6y-3=0$
• C. $x^2+y^2+x+6y+3=0$
• D. $x^2+y^2+4x-3y+3=0$

Answer 1

The correct option is A $x^2+y^2+x-6y+3=0$

REF.Image.

Let's consider given circle

$S_1=0$ &

equation of required circle

$as{S}_2=0$

$E{q}^n$ of tangent at P(2,3) to $S_1=0$

is $x=2$

(because equation of CP is y = 3 and tangent at P is ${\perp }^{le}$ to

Required circle should touch $S_1=0$ at $P(2,3)$

$\Rightarrow$ tangent at $P(2,3)$ to $S_1=0$ should be

common tangent for $S_1=0\&S_2=0$

$\Rightarrow$ Required circle should touch the line x = 2

at P(2,3)

$e{q}^n$ of $S_2=0$ will belong to family of circles

$S+\lambda L=0$ [ S is point circle of L is tangent at p]

i.e.$(x-2{)}^2+(y-3{)}^2+\lambda (x-2)=0$

It should satisfy (1,1) $\Rightarrow (1-2{)}^2+(1-3{)}^2+\lambda (1-2)=0$

$\Rightarrow \lambda =5$

equation of required circle

$(x-2{)}^2+(y-3{)}^2+5(x-2)=0$

$\Rightarrow x^2+y^2+x-6y+3=0$