Question

Assertion :The circles $x^2+y^2+2ax+c=0$ & $x^2+y^2+2by+c=0$ touch each other if $\frac{1}{c}=\frac{1}{a^2}+\frac{1}{b^2}$. Reason: Two cirlces with centre $c_1,c_2$ & radii $r_1,r_2$ touch each other if $c_1+c_2=r_1\pm r_2$.

• 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

Two circles touch each other $c_1{c}_2=r_1\pm r_2$
$\therefore \sqrt{a^2+b^2}=\sqrt{a^2-c}\pm \sqrt{b^2-c}$
$\Rightarrow a^2+b^2=a^2-c+b^2-c\pm 2\sqrt{(a^2-c)(b^2-c)}$
$\Rightarrow c^2=(a^2-c)(b^2-c)$
$\Rightarrow c^2=a^2{b}^2-c(a^2+b^2)+c^2$
$\Rightarrow c(a^2+b^2)=a^2{b}^2$
$\Rightarrow \frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c}$
Assertion (A) '& Reason (R) both are true & Reason
(R) is correct explanation of Assertion (A)
Ans: A