First slide

Parabola

Question

The focus of the parabola $4 y^{2} + 12 x - 20 y + 67 = 0$

Moderate

A

$(- 7 / 2, 5 / 2)$
B

$(- 3 / 4, 5 / 2)$
C

$(- 17 / 4, 5 / 2)$
D

$(5 / 2, - 3 / 4)$

Solution

The given equation of the parabola can be written as
$y^{2} - 5 y = - 3 x - 67 / 4 \Rightarrow (y - 5 / 2)^{2} = - 3 (x + 7 / 2)$
$\Rightarrow Y^{2} = 4 a X$ where $Y = y - 5 / 2, X = x + 7 / 2$ and $a = - 3 / 4$
The focus of $Y^{2} = 4 a X is (X, Y) = (a, 0) = (- 3 / 4, 0)$
$\Rightarrow x + 7 / 2 = - 3 / 4, y - 5 / 2 = 0 \Rightarrow x = - 17 / 4, y = 5 / 2$
Therefore, required focus is $(- 17 / 4, 5 / 2)$

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