A) the x-axis
B) the tangent at the vertex
C) the directrix
D) None of the above
Correct Answer: B
Solution :
Let S(a, 0) be the focus of the parabola\[{{y}^{2}}=4ax\] and\[P(a{{t}^{2}},\,\,2at)\]be a point on it. Then, the equation of a circle on SP as diameter is\[(x-a)(x-a{{t}^{2}})+(y-0)(y-2at)=0\] It meets y-axis at\[x=0\] \[\therefore \] \[{{y}^{2}}-2aty+{{a}^{2}}{{t}^{2}}=0\] \[\Rightarrow \] \[{{(y-at)}^{2}}=0\] This shows that y-axis meets the circle in two coincident points. Hence, the circle touches the tangent at the vertex.
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