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Pair of Straight Lines — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsPair of Straight Lines3 Marks
Question
If angle between the lines represented by $a x^2+2 h x y+b y^2=0$ is equal to the angle between the lines represented by $2 x^2-5 x y+3 y^2=0$, then show that $100\left(h^2-a b\right)=(a+b)^2$.
✓

Answer

Let $\theta$ and $\alpha$ be the angles between the pairs of lines
$a x^2+2 h x y+b y^2=0....(1)$ and
$2 x^2-5 x y+3 y^2=0....(2)$ respectively
In equation $(2)$
$a=2, b=3, h=-\frac{5}{2}$
$\therefore \tan \theta=\left|\frac{2 \sqrt{h^2-a b}}{a+b}\right|$
and $\tan \alpha=\left|\frac{2 \sqrt{\frac{25}{4}-6}}{5}\right|$
Given that the angle between the lines $(1)$ and
$(2)$ are equal.
$\therefore \theta=\alpha$
$\therefore\left|\frac{2 \sqrt{h^2-a b}}{a+b}\right|=\left|\frac{2 \sqrt{\frac{25}{4}-6}}{5}\right|$
$\therefore\left|\frac{\sqrt{h^2-a b}}{a+b}\right|=\frac{1}{10}$
Squaring both sides,
$\frac{h^2-a b}{(a+b)^2}=\frac{1}{100}$
$\therefore 100\left(h^2-a b\right)=(a+b)^2$
Hence proved

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