Question

Assertion :The spheres $x^2+y^2+z^2+2mx+c=0$ $x^2+y^2+z^2+2ny+c=0$ touches if $\frac{1}{m^2}+\frac{1}{n^2}=\frac{1}{c}$ Reason: Two spheres with centre $C_1$ and $C_2$ and radii $r_1,r_2$ touches each other if $|r_1\pm r_2|=\left|C_1{C}_2\right|$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

$c_1=(-m,0,0)$ & $r_1=\sqrt{m^2-c}$
$Also{c}_2=(0,-n,0),r_2=\sqrt{n^2-c}$
$\therefore \left|c_1{c}_2\right|=\sqrt{{(-m)}^2+n^2}=\sqrt{m^2+n^2}$
Spheres touch each other
$\therefore \left|c_1{c}_2\right|=|r_1\pm r_2|$
$\Rightarrow \sqrt{m^2+n^2}=\sqrt{m^2-c}\pm \sqrt{n^2-c}$
On squaring,we get
$\Rightarrow m^2+n^2=m^2-c+n^2-c\pm 2\sqrt{(m^2-c)}\sqrt{n^2-c}$
$\Rightarrow c^2=(m^2-c)(n^2-c)$
$\Rightarrow c^2=m^2{n}^2-c\left(m^2{n}^2\right)+c^2$
$\Rightarrow \frac{1}{c}=\frac{m^2+n^2}{m^2{n}^2}$
$\Rightarrow \frac{1}{c}=\frac{1}{m^2}+\frac{1}{n^2}$
$\Rightarrow$ Assertion(A) is correct as well as Reason (R)is so with correct explanation for Assertion (A).