A) \[x+a=0\]
B) \[2x+a=0\]
C) \[x=0\]
D) \[x=\frac{a}{2}\]
Correct Answer: C
Solution :
Let the coordinates of focus be \[S(a,0)\] Let any point on the parabola be \[P(a{{t}^{2}},2at).\]Let the coordinates of mid point of P and S be \[({{x}_{1}},{{y}_{1}}).\] \[\therefore \] \[{{x}_{1}}=\frac{a+a{{t}^{2}}}{2},{{y}_{1}}=\frac{0+2at}{2}\] \[\Rightarrow \] \[a{{t}^{2}}=2{{x}_{1}}-{{a}_{1}},\] \[{{y}_{1}}=at\] \[\Rightarrow \] \[a{{\left( \frac{{{y}_{1}}}{a} \right)}^{2}}=2{{x}_{1}}-a\] \[\Rightarrow \] \[y_{1}^{2}=2{{x}_{1}}a-{{a}^{2}}\] Hence, the locus of the mid point is \[{{y}^{2}}=2a\left( x-\frac{a}{2} \right)\] \[\therefore \] Equation of directrix is \[x-\frac{a}{2}=-\frac{a}{2}\] ie, \[x=0\]
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