If the point $P (α, β),$ is on the curve $x^{2} = 2 y,$ which is nearest to the point $Q (0, 5),$ then $α^{2} + β^{2}$ is

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Solution

$P (α, β)$ is on $x^{2} = 2 y$
$\Rightarrow α^{2} = 2 β$
Let $f = P Q^{2} = α^{2} + (β - 5)^{2} = β^{2} - 8 β + 25$
$\frac{d f}{d β} = 2 β - 8$ and $\frac{d^{2} f}{d β^{2}} = 2 > 0$
$∴ f$ will be minimum when $\frac{d f}{d β} = 0$
$\Rightarrow β = 4$
$∴ α^{2} = 8$
$\Rightarrow α^{2} + β^{2} = 24$

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