Assertion :The locus of the center of the circle described on any focal chord of a parabola $y^{2} = 4 a x$ as diameter is $y^{2} = 2 a (x - a)$ Reason: If $A (a t_{1}^{2}, 2 a t_{1})$ and $B (a t_{2}^{2}, 2 a t_{2})$ be the extremities of a focal chord for the parabola $y^{2} = 4 a x,$ then $t_{1} t_{2} = - 1$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
If $A ({a t}_{1}^{2}, {a t}_{1}), B ({a t}_{2}^{2}, 2 {a t}_{2})$ be the extremities of a focal chord for the parabola $y^{2} = 4 a x$ , then $t_{2} t_{2} = - 1.$
we want to find locus of point $C (α, β)$ where $C$ is the centre of the circle having $A B$ as the diameter.
$\Rightarrow α = \frac{a}{2} (t_{2}^{2} + t_{2}^{2}); β = a (t_{1} + t_{2})$
To eliminate $t_{1}, t_{2}$
$β^{2} = a^{2} (t_{1}^{2} + t_{2}^{2} + 2 t_{1} t_{2}) = a^{2} (\frac{2 α}{a} - 2) \Rightarrow β^{2} = 2 a (α - a)$ i.e., $y^{2} = 2 a (x - a)$

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