Question

If the plane 2ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres and $\begin{array}{c}{x}^2+y^2+z^2+6x-8y-2z=13\\ x^2+y^2+z^2-10x+4y-2z=8\end{array}$, then a equals
[AIEEE 2005]

• A. -2
• B. 2
• C. -1
• D. 1

Answer 1

The correct option is A

-2

$S_1\equiv x^2+y^2+z^2+6x-8y-2z=13,C_1\equiv (-3,4,1)$
$S_2\equiv x^2+y^2+z^2-10x+4y-2z=8,C_2\equiv (5,-2,1)$
So mid point of $C_1{C}_2$ (say P)
$\equiv P(\frac{5-3}{2},\frac{4-2}{2},\frac{1+1}{2})=P(1,1,1)$
Now the plane 2ax-3ay+4az+6=0 passes through the point P,
So, 2a(1)-3a(1)+4a(1)+6=0=2a-3a+4a+6=0
$\Rightarrow$ 3a+6=0 $\Rightarrow$ 3a=-6 $\Rightarrow$ a=-2