HyperbolaHardBloom L3
Question
The locus of the foot of the perpendicular from the centre of the hyperbola $xy = c^{2}$ on a variable tangent is:
Options
A.$\left( x^{2} - y^{2} \right)^{2} = 4c^{2}xy$
B.$\left( x^{2} + y^{2} \right)^{2} = 2c^{2}xy$
C.$\left( x^{2} + y^{2} \right) = 4c^{2}xy$
D.$\left( x^{2} + y^{2} \right)^{2} = 4c^{2}xy$
Solution
Sol. (d) Equation of tangent is
$$\begin{array}{r} hx + xy = h^{2} + k^{2}\#(1) \end{array}$$
Also, tangent at $P$ is
$$\begin{array}{r} \frac{x}{t} + yt = 2c\#(2) \end{array}$$
∵ (1) and (2) are identical
$${\Rightarrow ht = \frac{k}{t} = \frac{h^{2} + k^{2}}{2c} }{\Rightarrow \frac{h^{2} + k^{2}}{2ch} = \frac{2ck}{h^{2} + k^{2}} }{\Rightarrow \left( x^{2} + y^{2} \right)^{2} = 4c^{2}xy}$$
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