Question

The combined equation of the pair of lines passing through origin which are at a distance $4$ units from the point $(5,6)$ is

• A. $9x^2+60xy-20y^2=0$
• B. $9x^2-60xy+20y^2=0$
• C. $20x^2+60xy-9y^2=0$
• D. $20x^2-60xy+9y^2=0$

Answer 1

Answer: (D)

$20x^2-60xy+9y^2=0$

Let, $y=m_{1}x$ & $y=m_{2}x$ are the lines passing through origin and at $4$ units distance from $(5,6).$
$\therefore 4=\left|\frac{6-5m_1}{\sqrt{1+m_{1^2}}}\right|$ & $\left|\frac{6-5m_2}{\sqrt{1+m_{2^2}}}\right|$ ...(1)
From (1),
$16+16m_1^2=36+25m_1^2-60m_1$
$9m_1^2-60m_1+20=0$
$m_1=\frac{60\pm \sqrt{2880}}{18}$

$=\frac{60\pm 2\sqrt{720}}{18}=\frac{60\pm 24\sqrt{5}}{18}$

$m_1=\frac{10\pm 4\sqrt{5}}{3}$

$\therefore y^2-(m_1+m_2)xy+m_1{m}_2{x}^2=0$
$y^2-(2\times \frac{10}{3})xy+\left(\frac{100-80}{9}\right)x^2=0$

$\Rightarrow 9y^2-60xy+20x^2=0$

Hence, option D is the correct answer.