Question

If the sum of the slopes of the lines given by $x^2-2cxy-7y^2=0$ is four times their product, then $c$ has the value:

• A. $-2$
• B. $-1$
• C. $2$
• D. $1$

Answer 1

The correct option is C

$2$

Explanation for the correct option.

Let the slopes of the given lines $x^2-2cxy-7y^2=0$ be $m_1\mathrm{and}{m}_2$.

The standard equation of pair of lines is $a{x}^2+2hxy+b{y}^2=0$

By comparing the given equation, with the standard equation, we get $a=1,h=-c,\mathrm{and}b=-7$.

The sum of slopes is $-\frac{2h}{b}=-\frac{2c}{7}$

The product of slopes is $\frac{a}{b}=-\frac{1}{7}$

As per the given information,

$\begin{array}{rcl}-\frac{2c}{7}& =& 4\left(-\frac{1}{7}\right)\\ & \Rightarrow & -\frac{2c}{7}=-\frac{4}{7}\\ \Rightarrow 2c& =& 4\\ \Rightarrow c& =& 2\end{array}$

Hence, option C is correct.