Let P be the parabola,whose focus is (-2, 1) | vedclass

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(A) The definition of a parabola is the locus of a point $P(x, y)$ such that its distance from the focus is equal to its perpendicular distance from t

Vedclass · Accepted Answer

$\frac{3}{2}$

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(A) The definition of a parabola is the locus of a point $P(x, y)$ such that its distance from the focus is equal to its perpendicular distance from the directrix.Given focus $S = (-2, 1)$ and directrix $L: 2x + y + 2 = 0$.The equation of the parabola is $(x + 2)^2 + (y - 1)^2 = \left(\frac{2x + y + 2}{\sqrt{2^2 + 1^2}}\right)^2$.$5[(x + 2)^2 + (y - 1)^2] = (2x + y + 2)^2$.To find the ordinates of the points on the parabola with abscissa $x = -2$,substitute $x = -2$ into the equation:$5[(-2 + 2)^2 + (y - 1)^2] = (2(-2) + y + 2)^2$.$5(y - 1)^2 = (y - 2)^2$.$5(y^2 - 2y + 1) = y^2 - 4y + 4$.$5y^2 - 10y + 5 = y^2 - 4y + 4$.$4y^2 - 6y + 1 = 0$.This is a quadratic equation in $y$. Let the roots be $y_1$ and $y_2$. The sum of the ordinates is the sum of the roots,which is given by $-\frac{b}{a} = -\frac{-6}{4} = \frac{6}{4} = \frac{3}{2}$.

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

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