Question

If the slopes of one of the lines given by $a{x}^2+2hxy+b{y}^2=0$ is $5$ times the other, then

• A. $5h^2=9ab$
• B. $5h^2=ab$
• C. $h^2=ab$
• D. $9h^2=5ab$

Answer 1

The correct option is A

$5h^2=9ab$

Explanation of correct option.

Step 1: Use the relation between slopes and coefficient of a pair of lines represented by a second-degree equation

Let the slope of one of the line represented by the equation $a{x}^2+2hxy+b{y}^2=0$ be $m$, then the slope of another line will be $5m$

We know that for the pairs of the equation represented by $a{x}^2+2hxy+b{y}^2=0$,

The product of the slopes of two lines $=\frac{a}{b}$

The sum of the slopes of the two lines $=-\frac{2h}{b}$

But the slopes of two lines are $m$ and $5m$

$\therefore m\left(5m\right)=\frac{a}{b}....\left(1\right)$ and $m+5m=-\frac{2h}{b}....\left(2\right)$

Step 2: Eliminate $m$ from both equations $\left(1\right)$ and $\left(2\right)$

From $\left(2\right)$, we have

$$\begin{gathered} \Rightarrow m+5m=-\frac{2h}{b} \\ \Rightarrow 6m=-\frac{2h}{b} \\ \Rightarrow m=-\frac{h}{3b}\dots \text{(3)} \end{gathered}$$

From $\left(1\right)$, we have

$$\begin{gathered} m\left(5m\right)=\frac{a}{b} \\ \Rightarrow 5m^2=\frac{a}{b} \\ \Rightarrow m^2=\frac{a}{5b}....\left(4\right) \end{gathered}$$

Squaring $\left(3\right)$ and equating to $\left(4\right)$, we get

$\frac{h^2}{9b^2}=\frac{a}{5b}$

$\Rightarrow 5h^2=9ab$

Hence, Option (A) is the correct answer.