A parabola $y^{2} = 16 x$ translated to the coordinates $(1, 4) .$ If the equation of translated parabola is $y^{2} + a x + b y + c = 0.$ Find the value of $(a + b + c)$ .
8

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Solution

The correct option is A 8
Given equation of parabola is $y^{2} = 16 x .$
This parabola is translated to $(1, 4) .$

The equation of parabola which is translated to the point $(h, k)$ is given by $(y - k)^{2} = 4 a (x - h) .$

So, according to question we get the equation of parabola as $(y - 4)^{2} = 16 (x - 1)$

On simplifying we get,
$\Rightarrow y^{2} - 8 y + 16 = 16 x - 16$
$\Rightarrow y^{2} - 16 x - 8 y + 32 = 0$

On comparing with the given equation $y^{2} + a x + b y + c = 0$ we get, $a = - 16, b = - 8, c = 32.$
Hence, $a + b + c = - 16 - 8 + 32 = 8.$

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