Question

Assertion :The circle described on the line segment joining the points $(-2,-1)$ & $(0,-3)$ as diameter cuts the circle $x^2+y^2+5x+y+4=0$ orthogonally. Reason: $(-2,-1)$ & $(0,-3)$ are conjugate points w.r.t to the circle $x^2+y^2+5x+y+4=0$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Equation of circle whose end points of diameter are $(-2,-1)$ & $(0,-3)$ is $(x+2)(x-0)+(y+1)(y+3)=0$ $\Rightarrow x^2+y^2+2x+4y+3=0$ ....(1)
Now equation (1) and $x^2+y^2+5x+y+4=0$ are orthogonal when $2g_1{g}_2+2f_1{f}_2=c_1+c_2$
Also the points $(-2,-1)$ & $(0,-3)$ are conjugate w.r.t to the circle
$x^2+y^2+5x+y+4=0($ since $S_{12}=0)$ $\therefore$
Assertion is true and reason is so. But reason is the correct explanation of assertion.
Since, if the points $P$ and $Q$ are conjugate w.r.t to be circle $S=0,$ then the circle describe on $PQ$ as diameter cuts the circle $S=0$ orthogonally.