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Four Common Forms of Parabola Equation

The line 2x...

Question

The line $2 x + 3 y + 9 = 0$ touches the parabola $y^{2} = 8 x$ at the point

(0, - 3)

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(2, 4)

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(- 6, \frac{9}{2})

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(\frac{9}{2}, - 6)

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Solution

The correct option is D $(\frac{9}{2}, - 6)$
Given lines and parabola are $2 x + 3 y + 9 = 0$ and $y^{2} = 8 x$ respectively
To get the point of intersection, solve both the curves
$y^{2} = 4 (2 x) = 4 (- 3 y - 9)$ , use line equation
$\Rightarrow y^{2} + 12 y + 36 = 0$
$\Rightarrow (y + 6)^{2} = 0$
$\Rightarrow y = - 6$
Thus $x = - \frac{1}{2} (3 y + 9) = - \frac{1}{2} (- 18 + 9) = \frac{9}{2}$

Hence the point of intersection is $(\frac{9}{2}, - 6)$

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