Question

If the equations $k{x}^2-2xy-y^2-2x+2y=0$represents a pair of lines, then $k$ is equal to

• A. $2$
• B. $-2$
• C. $-5$
• D. $3$

Answer 1

The correct option is D

$3$

Explanation for correct option

Step 1: Comparing with the general equation of pair of straight line

Given equation is $k{x}^2-2xy-y^2-2x+2y=0$

The general equation of the pair of straight line is $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

Now comparing with the general equation of the straight line we get

$a=k,h=-1,b=-1,g=-1,f=1,c=0$

Step 2: Find the condition

The necessary and sufficient condition for general equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represent a pair of straight line is

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

$h^2>ab$

$g^2>ac$

From the first condition we have,

$$\begin{gathered} abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0 \\ \Rightarrow 0+2-k+1-0=0 \\ \Rightarrow k=3 \end{gathered}$$

Hence, Option (D) is correct i.e. $3$