First slide

Equation of line in Straight lines

Question

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

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

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

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