First slide

Parabola

Question

If perpendiculars are drawn on any tangent to a
parabola $y^{2} = 4 a x$ from the points $(a \pm k, 0)$ on the
axis. The difference of the squares of their lengths
is

Moderate

A

4
B

$4 a$
C

$4 k$
D

$4 a k$

Solution

Any tangent is $y = m x + a / m$ Required difference
is
$\begin{array}{l} {[\frac{m (a + k) + a / m}{\sqrt{1 + m^{2}}}]}^{2} - {[\frac{m (a - k) + a / m}{\sqrt{1 + m^{2}}}]}^{2} \\ = \frac{1}{1 + m^{2}} \times 4 (m a + a / m) m k = 4 a k \end{array}$

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