First slide

Parabola

Question

chord is drawn through the focus of the parabola
$y^{2} = 6 x$ such that 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

Let the equation of the chord be.
$y - 0 = m (x - \frac{3}{2}) \Rightarrow y = m x - \frac{3 m}{2}$
Its distance from the vertex (0, 0) is $|\frac{c}{\sqrt{a^{2} + b^{2}}}| = |\frac{m (- \frac{3}{2})}{\sqrt{1 + m^{2}}}| = \frac{\sqrt{5}}{2}$
$\frac{3 m}{\sqrt{1 + m^{2}}} = \sqrt{5}$
squaring on both sides
$9 m^{2} = 5 (1 + m^{2}) 4 m^{2} = 5 m = \pm \frac{\sqrt{5}}{2}$
.

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