First slide

Parabola

Question

A chord is drawn through the focus of the parabola $y^{2} = 6 x$ such than its distance from the vertex of this parabola is
$\frac{\sqrt{5}}{2}$ , then its slope can be

Moderate

A

$\frac{\sqrt{5}}{2}$
B

$\frac{\sqrt{3}}{2}$
C

$\frac{2}{\sqrt{5}}$
D

$\frac{2}{\sqrt{3}}$

Solution

Focus of the parabola is $(\frac{3}{2}, 0)$ Let the equation of the chord be
$y = m (x - \frac{3}{2})$
Its distance from the vertex $(0, 0)$ is $|\frac{- \frac{3 m}{2}}{\sqrt{1 + m^{2}}}| = \frac{\sqrt{5}}{2}$
$\Rightarrow 9 m^{2} = 5 (1 + m^{2})$
$\Rightarrow m^{2} = \frac{5}{4} \Rightarrow m = \frac{\sqrt{5}}{2}$

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