Question

The equation of the smallest circle passing thorough intersecting of line $x+y=1$the circle $x^2+y^2=9$, is:

• A. $x^2+y^2$=$9+x+y-8=0$
• B. $x^2+y^2$=$9-x-y-8=0$
• C. $x^2+y^2$=$9-x+y-8=0$
• D. none of these

Answer 1

The correct option is A $x^2+y^2$=$9+x+y-8=0$

We have,

The equation of family of circle passing through the point of $S+\lambda L=0$

$x^2+y^2-9-\lambda (x+y-1).......\left(1\right)$

Where

$S=x^2+y^2-9$

$L=x+y-1$

We observe that, line $y=x$ symmetric to both circle and line.

Then mid point will be point of intersection of $y=x$ and $x+y=1$

Which

$x=\frac{1}{2},y=\frac{1}{2}$

Put the value of $x$ and $y$ in equation (1) and we get,

$\lambda =1$

Then,

$x^2+y^2-9-1(x+y-1)=0$

$\Rightarrow x^2+y^2-9-x-y+8=0$

$\Rightarrow x^2+y^2=9+x+y-8$

Hence, this is the answer.