Find the coordinates of a point on the parabola `y=x^2+7x+2` which is closest to the straight line `y=3x-3.`

Explanation

Let the coordinates of the required point of the parabola `y=x^2+7 x+2` be `(h, k)`.

`therefore k=h^2+7 h+2 quad ldots . .(1)`

Distance of the point `(h, k)` from the straight line `y=3 x-3` is given by `D=(|3 h-k-3|)/(sqrt(3)^2)+(-1)^2=(|3 h-k-3|)/sqrt(10) ldots ldots .(2)`

From (1) and (2), we get

`D=(| 3 h-h^2-7 h-2-3|)/sqrt(10)`

`=(|-h^2-4 h-5|)/sqrt(10) `

`=(h^2+4 h+5)/sqrt(10)`

Differentiating both sides w.r.t. `h_t`, we get

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