Q70 / 160 · TS EAMCET 2003 · Mathematics
The equation of the parabola with focus $(0,0)$ and directrix $x+y=4$ is
- Correct answerA$x^2+y^2-2 x y+8 x+8 y-16=0$
- B$x^2+y^2-2 x y+8 x+8 y=0$
- C$x^2+y^2+8 x+8 y-16=0$
- D$x^2-y^2+8 x+8 y-16=0$
Solution
Given focus of parabola is $S(0,0)$.
Equation of directrix is $x+y=4$
Let $P(x, y)$ be any point on the parabola.
Now, $S P^2=P M^2$
$\Rightarrow \quad(x-0)^2+(y-0)^2=\left[\frac{x+y-4}{\sqrt{1+1}}\right]^2$
$\Rightarrow \quad x^2+y^2=\frac{(x+y-4)^2}{2}$
$\Rightarrow \quad 2 x^2+2 y^2=x^2+y^2+16+2 x y-8 x-8 y$
$\therefore \quad x^2+y^2-2 x y+8 x+8 y-16=0$
Hence, the correct answer is option: 'A'.