The base BC of a triangle ABC is bisected at the point (a, b) and equation to the sides AB and AC are respectively ax + by = 1 and bx + ay = 1. Equation of the median through A is :

a x - b y = a b

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(2 b - 1) (a x + b y) = a b

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(2 a b - 1) (a x + b y - 1) = (a^{2} + b^{2} - 1) (b x + a y - 1)

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b x - a y = 1

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Solution

The correct option is A $a x - b y = a b$
Equation of line passing through AB
$(a x + b y - 1) + λ (b x + a y - 1) = 0$
It passes through (a, b)
$∴ (a^{2} + b^{2} - 1) + λ (2 a b - 1) = 0$
$\Rightarrow λ = (\frac{1 - a^{2} - b^{2}}{2 a b - 1})$
Substitute $λ$ in
(i) $(a x + b y - 1) + (\frac{1 - a^{2} - b^{2}}{2 a b - 1}) (b x + a y - 1) =$
$\Rightarrow (a x + b y - 1) = \frac{(a^{2} + b^{2} - 1)}{(2 a b - 1)} (b x + a y - 1)$
$\Rightarrow (2 a b - 1) (a x + b y - 1) = (a^{2} + b^{2} - 1) (b x + a)$
Thus, option $C$ is correct

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