MCQ
The equation of the parabola whose focus is (1, -1) and the directrix is x + y + 7 = 0 is
- A$x^2+y^2-2 x y-18 x-10 y=0$
- B$x^2-18 x-10 y-45=0$
- C$x^2+y^2-18 x-10 y-45=0$
- ✓$x^2+y^2-2 x y-18 x-10 y-45=0$
✓
Answer
Correct option: D.
$x^2+y^2-2 x y-18 x-10 y-45=0$
- $x^2+y^2-2 x y-18 x-10 y-45=0$
Let P (x, y) be any point on the parabola whose focus is S (1, -1) and the directrix is x + y+ 7 = 0.
Draw PM perpendicular to x + y + 7 = 0.
Then, we have:
SP = PM
$\Rightarrow SP^2 = PM^2$
$\Rightarrow\ (\text{x} - 1)^2+ (\text{y} + 1)^2= \Big(\frac{\text{x+y+7}}{\sqrt{1+1}}\Big)^2$
$\Rightarrow\ (\text{x} - 1)^2+ (\text{y} + 1)^2= \Big(\frac{\text{x+y+7}}{\sqrt{2}}\Big)^2$
$\Rightarrow\ 2(\text{x}^2+1-2\text{x}+\text{y}^2+1+2\text{y})\\ \ \ =\text{x}^2+\text{y}^2+49+2\text{xy}+14\text{y}+14\text{x}$
$\Rightarrow\ (2\text{x}^2+2-4\text{x}+2\text{y}^2+2+4\text{y})\\ \ \ =\text{x}^2+\text{y}^2+49+2\text{xy}+14\text{y}+14\text{x}$
$\Rightarrow\ \text{x}^2+\text{y}^2-45-10\text{y}-2\text{xy}-18\text{x}=0$
Hence, the required equation is $x^2 + y^2 - 2xy - 18x - 10y - 45 = 0$.
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