First slide

Parabola

Question

The point of contact of the tangent to the parabola
$y^{2} = 9 x$ which passes through the point (4, 10) and
makes an angle $θ$ with the axis of the parabola such
that $\tan θ > 2$ is

Moderate

A

(4/9, 2)
B

(36, 18)
C

(4, 6)
D

(1/4, 3/2)

Solution

Equation of the tangent at any point $(9 t^{2} / 4, 9 t / 2)$
on the parabola is $t y = x + 9 t^{2} / 4$ . Since it passes
through (4, 10), $10 t = 4 + 9 t^{2} / 4 or 9 t^{2} - 40 t + 16$
=0.
$\Rightarrow t = 4 / 9, 4 \cdot \tan θ = 1 / t > 2 \Rightarrow t < 1 / 2$ . So
$t = 4 / 9$ and the required point is (4/9, 2).

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