First slide

Pair of straight lines

Question

The equation of a line which is parallel to the line common to the pair of lines given by $6 x^{2} - xy - 12 y^{2} = 0$ and $15 x^{2} + 14 xy - 8 y^{2} = 0$ and at a distance of 7 units from it is

Moderate

A

$3 x - 4 y = - 35$
B

$5 x - 2 y = - 7$
C

$3 x + 4 y = 35$
D

$2 x - 3 y = 7$

Solution

We have $6 x^{2} - xy - 12 y^{2} = 0$
or $(2 x - 3 y) (3 x + 4 y) = 0$ _(1)
and $15 x^{2} + 14 xy - 8 y^{2} = 0$
or $(5 x - 2 y) (3 x + 4 y) = 0$ (2)
Equation of the line common to (1) and (2) is
$3 x + 4 y = 0$ ____(3)
Equation of any line parallel to (3) is $3 x + 4 y = k$ Since its distance (3) is 7,
we have $| \frac{k}{\sqrt{3^{2} + 4^{2}}} | = 7$
or $k = \pm 35$

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