First slide

Pair of straight lines

Question

If $\frac{x}{a} + \frac{y}{b} = 1$ intersects $5 x^{2} + 5 y^{2} + 5 b x + 5 a y - 9 a b = 0$ at P and Q, $∠ P O Q = π / 2$ then the relation between a and b is

Moderate

A

a = b
B

a = 2b or b = 2a
C

a = 3b or b = 3a
D

a + b = 5

Solution

Homogenising,
$5 x^{2} + 5 y^{2} + (5 b x + 5 a y)$ $(\frac{x}{a} + \frac{y}{b}) - 9 a b {(\frac{x}{a} + \frac{y}{b})}^{2} = 0$
coefficient of $x^{2}$ + coefficient of $y^{2}$
$\Rightarrow (5 + \frac{5 b}{a} - \frac{9 b}{a}) + (5 + \frac{5 a}{b} - \frac{9 a}{b}) = 0$
$\Rightarrow b (5 a - 4 b) + a (5 b - 4 a) = 0$
$\Rightarrow (a - 2 b) (2 a - b) = 0$

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