Let P be the parabola, whose focus is (–2, 1) and directrix is 2x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose abscissa is –2, is [2025]

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Solution

Q.
Let P be the parabola, whose focus is (–2, 1) and directrix is 2x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose abscissa is –2, is [2025]

1 $\frac{5}{2}$

2 $\frac{3}{2}$

3 $\frac{3}{4}$

4 $\frac{1}{4}$

Ans.
(2)

Equation of parabola.

${(x + 2)}^{2} + {(y - 1)}^{2} = {(\frac{2 x + y + 2}{\sqrt{5}})}^{2}$

$\Rightarrow 5 [{(x + 2)}^{2} + {(y - 1)}^{2}] = {(2 x + y + 2)}^{2}$ ... (i)

Put x = –2, in equation (i),

$5 {(y - 1)}^{2} = {(y - 2)}^{2}$

$\Rightarrow 5 (y^{2} - 2 y + 1) = y^{2} - 4 y + 4$

$\Rightarrow 4 y^{2} - 6 y + 1 = 0$

$∴$ The sum of ordinates, $y_{1} + y_{2} = \frac{3}{2}$ .

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