First slide

Parabola

Question

$From a point (\sin θ, \cos θ), if three normals can be drawn to the parabola y^{2} = 4 a x, then the value of a is$

Moderate

A

$(1 / 2, 1)$
B

$[- 1 / 2, 0)$
C

$[- 1 / 2, 1]$
D

$(- 1 / 2, 0) \cup (0, 1 / 2)$

Solution

$Point (\sin θ, \cos θ) lies on the circle x^{2} + y^{2} = 1$ $\forall θ \in R$
$Now, three normals can be drawn to the parabola y^{2} = 4 a x if$
$x = | 2 a | meets this circle.$
$Hence, we must have$
$\begin{array}{l} \cos θ > | 2 a | \\ or 0 < | 2 a | < 1 \\ or 0 < | a | < \frac{1}{2} \\ or a \in (- \frac{1}{2}, 0) \cup (0, \frac{1}{2}) \end{array}$

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