If each line of a pair of lines passing through origin is at a perpendicular distance of 4 units from the…
If each line of a pair of lines passing through origin is at a perpendicular distance of 4 units from the point \((3,4)\), then the equation of the pair of lines is
\(7 x^2+24 x y=0\)
\(7 y^2+24 x y=0\)
\(7 y^2-24 x y=0\)
\(7 x^2-24 x y=0\)
Solution
Let equation of line passes through origin having slope \(m\) is \(y-m x=0\), according to given information
\(\begin{gathered}
\frac{|4-3 m|}{\sqrt{1+m^2}}=4 \\
\Rightarrow 16+9 m^2-24 m=16+16 m^2 \\
\Rightarrow 7 m^2+24 m=0 \Rightarrow m=0 \text { or } m=-\frac{24}{7}
\end{gathered}\)
so combined equation of required lines
\(\begin{aligned}
& y\left(y+\frac{24}{7} x\right) =0 \\
\Rightarrow \quad & 7 y^2+24 x y =0
\end{aligned}\)
Hence, option (2) is correct.