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 \[({{x}_{1}},{{y}_{1}}).\]You need to login to perform this action.
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