The base BC of triangle ABC is bisected at the point $(p, q)$ and the equations to the sides AB and AC are $p x + q y = 1$ and $q x + p y = 1$ . The equation of the median through A is

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$(2 p q - 1) (p x + q y - 1) = (p^{2} + q^{2} - 1) (q x + p y - 1)$

$q x - p y = 0$

$\frac{x}{p} + \frac{y}{q} = 2$

$(p - 2 q) x + (q - 2 p) y = p^{2} + r^{2}$

answer is C.

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Detailed Solution

Equation of the line through AB and AC is
$(p x + q y - 1) + λ (q x + p y - 1) = 0_____(i)$
$∴$ It passes through (p,q), then
$(p^{2} + q^{2} - 1) + λ (p q + p q - 1) = 0$
$∴ λ = - \frac{(p^{2} + q^{2} - 1)}{(2 p q - 1)}$
$∴$ From equation (i),
$(p x + q y - 1) - \frac{(p^{2} + q^{2} - 1)}{(2 p q - 1)} (q x + p y - 1) = 0$
or $(2 p q - 1) (p x + q y - 1) = (p^{2} + q^{2} - 1) (q x + p y - 1)$