Name: Pair of Lines Parallel to the Pair of Lines
Uploaded: 2019-07-19
Duration: 3 min 40 s
Description: We know how to find the equation of a line parallel to a given line. But how would you find the equation of pair of lines parallel to given pair of lines?

Question

Find the value of

k,

if the equation

12 x^{2} + 7 x y - 12 y^{2} - x + 7 y + k = 0

represents a pair of straight lines. Further, find whether these lines are parallel or intersecting.

Accepted Answer

Given pair of straight lines is

$12 x^{2} + 7 x y - 12 y^{2} - x + 7 y + k = 0 . . . . (1)$

General equation is

$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 b y + c = 0 . . . . . (2)$

From equation $(1) and (2)$

$a = 12, 2 h = 7 \Rightarrow h = \frac{7}{2}, b = - 12, 2 g = - 1 \Rightarrow g = - \frac{1}{2}, 2 f = 7 \Rightarrow f = \frac{7}{2}, C = k$

Condition to represent a pair of straight lines is

$a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0 . . . . (3)$

Substitute the values in equation $(3)$

$\Rightarrow 12 (- 12) k + 2 (\frac{7}{2}) (\frac{- 1}{2}) \frac{7}{2} - 12 \times \frac{49}{4} + 12 (\frac{1}{4}) - k (\frac{49}{4}) = 0$

$\Rightarrow - 144 k - \frac{49}{4} - 147 + 3 - \frac{49 k}{4} = 0$

$\Rightarrow 144 k + \frac{49 k}{4} = - \frac{49}{4} - 147 + 3$

$\Rightarrow \frac{(576 + 49) k}{4} = \frac{- 49 - 147 \times 4 + 3 \times 4}{4}$

$\Rightarrow \frac{(576 + 49) k}{4} = \frac{- 49 - 576}{4}$

$\Rightarrow 625 k = - 625$

Therefore, the value of $k = - 1$

We have $a + b = 12 - 12 = 0$

If $a + b = 0,$ then the lines are perpendicular.

Hence, the lines are intersecting.

Find the value of $k,$ if the equation $12 x^{2} + 7 x y - 12 y^{2} - x + 7 y + k = 0$ represents a pair of straight lines. Further, find whether these lines are parallel or intersecting.

Important Questions on Two Dimensional Analytical Geometry

Learn and Prepare for any exam you want !

Find the value of k, if the equation 12x2+7xy−12y2−x+7y+k=0 represents a pair of straight lines. Further, find whether these lines are parallel or intersecting.

Important Questions on Two Dimensional Analytical Geometry

Learn and Prepare for any exam you want !

Find the value of $k,$ if the equation $12 x^{2} + 7 x y - 12 y^{2} - x + 7 y + k = 0$ represents a pair of straight lines. Further, find whether these lines are parallel or intersecting.