Question

If the chords of rectangular hyperbola \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\]touches the parabola \[{{y}^{2}}=4ax\]then the locus of their mid – points is
(a) \[{{x}^{2}}\left( y-a \right)={{y}^{3}}\]
(b) \[{{y}^{2}}\left( x-a \right)={{x}^{3}}\]
(c) \[x\left( y-a \right)=y\]
(d) \[y\left( x-a \right)=x\]

Answer 1

Hint: Let us take a rough figure that represents that given information as follows

Here the red line represents that hyperbola, blue line represents the parabola and green line represents that chord of hyperbola that touches the parabola.
We assume that \[P\left( h,k \right)\] is the mid – point of chord AB of hyperbola.
We have the equation of chord of \[S\equiv {{x}^{2}}-{{y}^{2}}-{{a}^{2}}=0\] having mid – point as \[\left( {{x}_{1}},{{y}_{1}} \right)\]
\[{{S}_{1}}={{S}_{11}}\]
Where, \[{{S}_{1}}=x{{x}_{1}}-y{{y}_{1}}-{{a}^{2}}\] and \[{{S}_{11}}={{x}_{1}}^{2}-{{y}_{1}}^{2}-{{a}^{2}}\]
We have the condition that if \[y=mx+c\] touches \[{{y}^{2}}=4ax\] then \[c=\dfrac{a}{m}\]
By using the above conditions we find the locus of point \[P\left( h,k \right)\]

Complete step by step answer:
We are given that the chord of rectangular hyperbola \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\]touches the parabola \[{{y}^{2}}=4ax\]
We are asked to find the locus of mid – points of the chords.
Let us assume that the mid – point of chord of rectangular hyperbola as \[P\left( h,k \right)\]
We know that the equation of chord of \[S\equiv {{x}^{2}}-{{y}^{2}}-{{a}^{2}}=0\] having mid – point as \[\left( {{x}_{1}},{{y}_{1}} \right)\]
\[{{S}_{1}}={{S}_{11}}\]
Where, \[{{S}_{1}}=x{{x}_{1}}+y{{y}_{1}}-{{a}^{2}}\] and \[{{S}_{11}}={{x}_{1}}^{2}-{{y}_{1}}^{2}-{{a}^{2}}\]
By using the above condition we get the equation of AB having mid – point \[P\left( h,k \right)\] as
\[\begin{align}
  & \Rightarrow hx-ky-{{a}^{2}}={{h}^{2}}-{{k}^{2}}-{{a}^{2}} \\
 & \Rightarrow hx-ky={{h}^{2}}-{{k}^{2}} \\
\end{align}\]
Now, let us convert the above equation in the form of general equation of line that is \[y=mx+c\] then we get
\[\begin{align}
  & \Rightarrow ky=hx-\left( {{h}^{2}}-{{k}^{2}} \right) \\
 & \Rightarrow y=\dfrac{h}{k}x-\left( \dfrac{{{h}^{2}}-{{k}^{2}}}{k} \right)......equation(i) \\
\end{align}\]
We are given that this chord touches the parabola.
We know that the condition that if \[y=mx+c\] touches \[{{y}^{2}}=4ax\] then \[c=\dfrac{a}{m}\]
By using the above condition to equation (i) and given parabola \[{{y}^{2}}=4ax\] then we get
\[\begin{align}
  & \Rightarrow -\dfrac{{{h}^{2}}-{{k}^{2}}}{k}=\dfrac{a}{\left( \dfrac{h}{k} \right)} \\
 & \Rightarrow {{k}^{2}}-{{h}^{2}}=\dfrac{a{{k}^{2}}}{h} \\
\end{align}\]
Now, by cross multiplying the terms in the above equation we get
\[\begin{align}
  & \Rightarrow h{{k}^{2}}-{{h}^{3}}=a{{k}^{2}} \\
 & \Rightarrow {{k}^{2}}\left( h-a \right)={{h}^{3}} \\
\end{align}\]
Therefore, we get the locus of mid – point by replacing \[\left( h,k \right)\] with \[\left( x,y \right)\] as follows
\[\therefore {{y}^{2}}\left( x-a \right)={{x}^{3}}\]

So, the correct answer is “Option b”.

Note: Students may make mistakes in taking the second condition that is tangent condition of parabola.
We have the condition that if \[y=mx+c\] touches \[{{y}^{2}}=4ax\] then \[c=\dfrac{a}{m}\]
Here we need to take care that the tangent equation should be in the form \[y=mx+c\] then only this condition will hold.
We have the equation of line that touches the parabola as
\[\Rightarrow hx-ky={{h}^{2}}-{{k}^{2}}\]
Here we cannot apply the condition directly.
We need to convert the above equation in to general line equation that is \[y=mx+c\] as
\[\Rightarrow y=\dfrac{h}{k}x-\left( \dfrac{{{h}^{2}}-{{k}^{2}}}{k} \right)\]
Now, we need to apply the tangent condition of parabola.