If the ratio of gradients of the lines represented by $a x^{2} + 2 h x y + b y^{2} = 0$ is $1 : 3,$ then the value of the ratio $h^{2} : a b$ is

$\frac{1}{3}$
$\frac{3}{4}$
$\frac{4}{3}$
$1$

Open in App

Solution

The correct option is C
$\frac{4}{3}$

Explanation for correct option:
Step 1: Use the relationship between the slopes of lines represented by homogeneous equations
Let $m_{1}$ and $m_{2}$ be the slopes of the lines given by $a x^{2} + 2 h x y + b y^{2} = 0$
We know that the sum of the slopes of lines represented by the equation $a x^{2} + 2 h x y + b y^{2} = 0$ is $\frac{- 2 h}{b}$ and their product is given by $\frac{a}{b}$ .
$∴ m_{1} + m_{2} = \frac{- 2 h}{b}$ $. . . . (1)$ and $m_{1} m_{2} = \frac{a}{b}$ $. . . . (2)$
Step 2: Eliminate $m_{1}$ and $m_{2}$ from $(1)$ and $(2)$
It is given that the ratio of the slope of the given lines is $1 : 3$
$∴ \frac{m_{1}}{m_{2}} = \frac{1}{3} \Rightarrow m_{2} = 3 m_{1}$
Substituting the value of $m_{2}$ in $(1)$ , we get
$4 m_{1} = \frac{- 2 h}{b}$
$\Rightarrow$ $m_{1} = \frac{- h}{2 b}$
Substituting the value of $m_{1}$ and $m_{2}$ in $(2)$ , we get
$3 {(\frac{- h}{2 b})}^{2} = \frac{a}{b}$
$\Rightarrow$ $\frac{3 h^{2}}{4 b^{2}} = \frac{a}{b}$
$\Rightarrow$ $\frac{h^{2}}{a b} = \frac{4}{3}$
$\Rightarrow$ $h^{2} : a b = \frac{4}{3}$
Thus, option (C) is correct option.

Suggest Corrections

Similar questions

If the ratio of gradients of the lines represented by $a x^{2} + 2 h x y + b y^{2} = 0$ is $1 : 3$ , then the value of the ratio $h^{2} : a b$ is

a x^{2} + 2 h x y + b y^{2} = 0

3 : 1

h^{2}

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

(\frac{b g - h f}{h^{2} - a b}, \frac{a f - g h}{h^{2} - a b})

a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0

(\sqrt{(\frac{f^{2} - b c}{h^{2} - a b})}, \sqrt{(\frac{g^{2} - a c}{h^{2} - a b})})

θ

a x^{2} + 2 h x y + b y^{2} = 0

tan θ = ∣ ∣ ∣ \frac{2 \sqrt{h^{2} - a b}}{a + b} ∣ ∣ ∣, a + b \neq 0

a x^{2} + 2 h x y + b y^{2} = 0

\frac{a + b}{h} + \frac{8 h^{2}}{a b}

Join BYJU'S Learning Program