Question

Let $x^{2}=3ax+2ay+3$ is a family of parabola where $a$ is parameter. Then the locus of the vertex of the parabola is

• A. $3x^{2}+4xy+9=0$
• B. $4x^{2}+3xy+9=0$
• C. $4x^{2}+9xy+3=0$
• D. none of these

Answer 1

Answer: (A)

Given equation of parabola $x^{2}=3ax+2ay+3$ where a is variable on solving $\left(x - \frac{3 a}{2}\right)^{2}=2a\left(y + \frac{12 + 9 a^{2}}{8 a}\right),$ , vertex $v\left(\frac{3 a}{2} , - \frac{\left(12 + 9 a^{2}\right)}{8 a}\right)$
Let $v\left(\right.h,k\left.\right)$
$h=\frac{3 a}{2},k=-\frac{\left(12 + 9 a^{2}\right)}{8 a}$
$3h^{2}+4hk+9=0$
So, locus is $3x^{2}+4xy+9=0$