Question

The combined equation to a pair of straight lines passing through the origin and inclined at an angles ${30}^0$ and ${60}^0$ respectively with $X$-axis

• A. $\sqrt{3}(x^2+y^2)=4xy$
• B. $4(x^2+y^2)=\sqrt{3}xy$
• C. $x^2+\sqrt{3}{y}^2-2xy=0$
• D. $x^2+3y^2-2xy=0$

Answer 1

Answer: (A)

$\sqrt{3}(x^2+y^2)=4xy$

Equation of lines passing through origin and having ${30}^{\circ }$ and ${60}^{\circ }$

Angles with x-axis are

$y=tan{30}^{\circ }x$ and
$y=tan{60}^{\circ }$

$y=\frac{x}{\sqrt{3}}$ and
$y=\sqrt{3}x$

So, pair of straight lines of $y=\frac{x}{\sqrt{3}}$ and $y=\sqrt{3}x$ is

$(y-\frac{x}{\sqrt{3}})(y-\sqrt{3}x)=y^2+x^2-\sqrt{3}xy-\frac{xy}{\sqrt{3}}=0$

$\Rightarrow \sqrt{3}(x^2+y^2)-4xy=0$

$\Rightarrow \sqrt{3}(x^2+y^2)=4xy$