question_answer

The tangent at the point \[({{x}_{1}},{{y}_{1}})\]to the parabola y2 = 4ax meets the parabola \[{{y}^{2}}=4a\](x+b) at Q and R, then the midpoint of QR is

A)  \[({{x}_{1}},{{y}_{1}})\]               

B)  \[({{x}_{1}}+b,{{y}_{1}})\]

C)  \[({{x}_{1}}+b,{{y}_{1}}+b)\]    

D)  \[({{x}_{1}}-b,{{y}_{1}}-b)\]

Correct Answer: A

Solution :

The equation of the tangent to y2 = 4ax at point \[({{x}_{1}},{{y}_{1}})\]is \[y{{y}_{1}}=2a(x+{{x}_{1}}).\] or      \[2ax-y{{y}_{1}}+2a{{x}_{1}}=0\]                        ...(i)  Let (h, k) be the mid point of Q.R. Then, the equation of QR is \[(\because T={{S}_{1}})\] \[\Rightarrow \]\[2ax-ky+{{k}^{2}}-2ah=0\]                      ?(ii) Clearly, Eqs. (i) and (ii) represent the same line. \[\therefore \]\[\frac{1}{1}=\frac{{{y}_{1}}}{k}=\frac{2a{{x}_{1}}}{{{k}^{2}}-2ah}\] \[\Rightarrow \]\[k={{y}_{1}}\]and\[{{k}^{2}}-2ah=2a{{x}_{1}}\] \[\therefore \]    \[y_{1}^{2}-2ah=2a{{x}_{1}}\] \[\Rightarrow \]            \[4a{{x}_{1}}-2ah=2a{{x}_{1}}\] \[\Rightarrow \]            \[h={{x}_{1}}\] Hence, the mid point of OR is \[({{x}_{1}},{{y}_{1}}).\]