A) \[(a{{t}_{1}}{{t}_{2}},a({{t}_{1}}+{{t}_{2}}))\]
B) \[(a({{t}_{1}}+{{t}_{2}}),a{{t}_{1}}{{t}_{2}})\]
C) \[(t_{1}^{2}t_{2}^{2},{{t}_{1}}+{{t}_{2}})\]
D) None of these
Correct Answer: A
Solution :
Equation of tangent to the parabola\[{{y}^{2}}=4ax\] at the point\[(at_{1}^{2},2a{{t}_{1}})\]is \[2a{{t}_{1}}=2a(x+at_{1}^{2})\] \[\Rightarrow \] \[y{{t}_{1}}=x+at_{1}^{2}\] ...(i) Similarly, equation of tangent at the point\[(at_{2}^{2},2a{{t}_{2}})\]is \[2ya{{t}_{2}}=2a(x+at_{2}^{2})\] \[\Rightarrow \] \[y{{t}_{2}}=x+at_{2}^{2}\] ...(ii) On solving Eqs. (i) and (ii), we get \[y=a({{t}_{1}}+{{t}_{2}}),x=a{{t}_{1}}{{t}_{2}}\] \[\therefore \]Intersection point\[[a{{t}_{1}}{{t}_{2}},a({{t}_{1}}+{{t}_{2}})]\]
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