The locus of the point whose ratio of distance from the origin to its distance from \((-2,-3)\) is \(5: 7\),…
The locus of the point whose ratio of distance from the origin to its distance from \((-2,-3)\) is \(5: 7\), is given by.........
\(24\left(x^2+y^2\right)-100 x-150 y-325=0\)
\(24\left(x^2+y^2\right)+100 x+150 y-325=0\)
\(24\left(x^2+y^2\right)-100 x+150 y+325=0\)
\(2 x^2+2 y^2=325\)
Solution
Let the point \(P(x, y)\), such that \(O P: A P=5: 7\), where \(O\) is the origin and \(A(-2,-3)\), so
\(\begin{aligned}
& \frac{\sqrt{x^2+y^2}}{\sqrt{(x+2)^2+(y+3)^2}}=\frac{5}{7} \\
\Rightarrow \quad & 49\left(x^2+y^2\right)=25\left[x^2+y^2+4 x+6 y+13\right] \\
\Rightarrow \quad & 24\left(x^2+y^2\right)-100 x-150 y-375=0
\end{aligned}\)
Hence, option (a) is correct.