Question

Consider the family of circles $x^2+y^2-2x-2\lambda y-8=0$ passing through two fixed points A and B. Then the distance between the points A and B, is $\underset{–}{}$

• A. 6
• B. $\sqrt{6}$
• C. $\sqrt{18}$
• D. 36

Answer 1

The correct option is A

6

Given equation of family of circles

$x^2+y^2-2x-2\lambda y-8=0$

After rearranging it

We get,

$(x^2+y^2-2x-8)-2\lambda y=0$

Which is of the form of $S+\lambda L=0$ of families of circles

All the circles passes through the point of intersection of circle $x^2+y^2-2x-8=0$ and $y=0$

$x^2+y^2-2x-8=0$ - - - - - - -(1)

$y=0$ - - - - - - -(2)

Solving equations (1) $\&$ (2)

$x^2-2x-8=0$

or $(x-4)(x+2)=0$

$x=4,x=-2$

Point $A(4,0)$

Point $B(-2,0)$

$AB=\sqrt{(4+2{)}^2+}=\sqrt{36}=6$

Distance between point A and B is 6.