Question

The gradient of one of the lines of $a{x}^2+2hxy+b{y}^2=0$ is twice that of the other then

• A. $h^2=ab$
• B. $h=a+b$
• C. $8h^2=9ab$
• D. $9h^2=8ab$

Answer 1

The correct option is C

$8h^2=9ab$

Explanation for the correct answer:

Find the relation between $a,b$ and $h$ for the given condition.

Assume that, $m_1$ and $m_2$ be the slope of the lines represented by $a{x}^2+2hxy+b{y}^2=0$.

We know that,

$m_1+m_2=-\frac{2h}{b}...1$

Also, $m_1·m_2=\frac{a}{b}...2$

Since, it is given that, the slope of one line is twice the slope of the other line.

That is, $m_1=2m_2$

So, equation $1$ becomes:

$$\begin{gathered} 3m_2=-\frac{2h}{b} \\ \Rightarrow m_2=-\frac{2h}{3b}...3 \end{gathered}$$

Also, equation $2$ becomes:

$2{\left(m_2\right)}^2=\frac{a}{b}...4$

From equation $3$ and $4$, we get

$$\begin{gathered} 2{\left(-\frac{2h}{3b}\right)}^2=\frac{a}{b} \\ \Rightarrow \frac{8h^2}{9b^2}=\frac{a}{b} \\ \Rightarrow 8h^2=9ab \end{gathered}$$

Therefore, the relation between $a,b$ and $h$ when the slope of one line is twice the slope of the other line is $8h^2=9ab$.

Hence, option $C$ is the correct answer.