A) \[-3{{a}^{2}}\]
B) \[3{{a}^{2}}\]
C) \[-4{{a}^{2}}\]
D) \[4{{a}^{2}}\]
E) \[2{{a}^{2}}\]
Correct Answer: A
Solution :
Let\[P(at_{1}^{2},2a{{t}_{1}}),Q(at_{2}^{2},2a{{t}_{2}})\]be a focal chord of the parabola\[{{y}^{2}}=4ax\]. Therefore, the tangents at P and Q meet at \[[a{{t}_{1}}{{t}_{2}},a({{t}_{1}}+{{t}_{2}})]\] \[\therefore \]\[{{x}_{1}}=-a\]and\[{{y}_{1}}=a({{t}_{1}}+{{t}_{2}})\]and normal at P and Q, meet at \[[2a+a(t_{1}^{2}+t_{2}^{2}-1),a({{t}_{1}}+{{t}_{2}})]\] \[\therefore \] \[{{x}_{2}}=2a+a(t_{1}^{2}+t_{2}^{2}-1)\] and \[{{y}_{2}}=a({{t}_{1}}+{{t}_{2}})\] \[\therefore \]\[{{x}_{1}}{{x}_{2}}+{{y}_{1}}{{y}_{2}}=-a[2a+a(t_{1}^{2}+t_{2}^{2}-1)]\] \[+{{a}^{2}}{{({{t}_{1}}+{{t}_{2}})}^{2}}\] \[=-3{{a}^{2}}\]
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