Q94 / 160 · TS EAMCET 2020 (14 Sep Shift 1) · Mathematics
If $S(a, b)$ is a fixed point and $P(\alpha, \beta)$ is such a variable point that $4\left[(x-a)^2+(y-b)^2\right]=(\alpha x+\beta y+7)^2$ represents a parabola, then the locus of $P(\alpha, \beta)$ is
- A$\beta^2=4 \alpha$
- Correct answerB$\alpha^2+\beta^2=4$
- C$\frac{\alpha^2}{4}+\frac{\beta^2}{2}=1$
- D$(\alpha+\beta)^2=4$
Solution
Equation of parabola is
$\begin{aligned}
& 4\left[(x-a)^2+(y-b)^2\right]=(\alpha x+\beta y+7)^2 \\
& 4\left[(x-a)^2+(y-b)^2\right]=\left(\frac{\alpha x+\beta y+7}{\sqrt{\alpha^2+\beta^2}}\right)^2\left(\alpha^2+\beta^2\right)
\end{aligned}$
$\therefore$ It is a parabola if $\alpha^2+\beta^2=4$
Hence, the correct answer is option: 'B'.