First slide

Parabola

Question

If on a given base BC [B(0, 0) and C(2,0)], a triangle is described such that the sum of the tangents of the base angles is 4, then the equation of the locus of the opposite vertex A is a parabola whose directrix is, y = k. The value of k is

Moderate

Solution

Given $\tan α \cdot \tan β = 4$
$\begin{array}{l} or \frac{y}{x} + \frac{y}{2 - x} = 4 \\ or y = 2 x (2 - x) \\ or - \frac{y}{2} = x^{2} - 2 x = (x - 1)^{2} - 1 \\ or (x - 1)^{2} = - \frac{1}{2} (y - 2) \end{array}$
Directrix : $y - 2 = \frac{1}{8} or y = \frac{17}{8}$

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