First slide

Pair of straight lines

Question

The lines joining the origin to the points of intersection of $3 x^{2} + λ x y - 4 x + 1 = 0$ and $12 x + y - 1 = 0$ are at right angles for

Moderate

A

$λ = - 4$
B

$λ = 4$
C

$λ = 7$
D

all values of $λ$

Solution

Equation of the lines joining the origin to the
points of intersection of the given lines is
$3 x^{2} + λ x y - 4 x (2 x + y) + 1 \cdot (2 x + y)^{2} = 0$
(Making the equation of the pair of lines homogeneous with
the help of the equation of the line)
$\Rightarrow x^{2} - λ x y - y^{2} = 0$
which are perpendicular for all values of l as the product
of the slopes is $- 1$

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