HyperbolaHardBloom L3
Question
Number of common tangents with finite slope to the curves $xy = c^{2}$ and $y^{2} = 4ax$ is:
Options
A.0
B.1
C.2
D.4
Solution
Sol. (b) Equation of $PQ$
$$\begin{array}{r} yt - x = at^{2}\#(1) \end{array}$$
$$\begin{array}{r} \frac{x}{t^{'}} + yt^{'} = 2c\#(2) \end{array}$$
∵ (1) and (2) are identical
$${\Rightarrow \ - t^{'} = \frac{t}{t^{'}} = \frac{at^{2}}{2c} }{\Rightarrow \frac{a\left( t^{'} \right)^{4}}{2c} = - t^{'} }{\Rightarrow \ \left( t^{'} \right)^{3} = - \frac{2c}{a} }$$⇒ Only one real value of $t^{'}$ exist.
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