First slide

Pair of straight lines

Question

All chords of the curve $3 x^{2} - y^{2} - 2 x + 4 y = 0$ , which subtend a right angle at the origin, pass through the fixed point. The square of its distance from the origin is

Moderate

Solution

Given curve is $3 x^{2} - y^{2} - 2 x + 4 y = 0$ …………..(1)
Let the equation of the chord $l x + m y = 1$ ………. (2)
Homogenising (1) with the help of (2)
we get $3 x^{2} - y^{2} - (2 x - 4 y) (l x + m y) = 0$
coefficient of x² + coefficient of y² = 0
$\Rightarrow$ $ℓ - 2 m = 1$ …………(3)
from (2) and (3), we get (x,y) = (1, -2)
square of distance from (1,-2) to (0,0)
$1^{2} + {(- 2)}^{2} = 5$

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