Given Curve is y² = 4x …… (1)

Let us assume the point on the curve which is nearest to the point (2, - 8) be (x, y)

The (x, y) satisfies the relation(1)

Let us find the distance(S) between the points (x, y) and (2, - 8)

We know that distance between two points (x₁,y₁) and (x₂,y₂) is .

⇒

⇒

⇒

Squaring on both sides we get,

⇒ S² = x² + y² - 4x + 16y + 68

From (1)

⇒

⇒

We know that distance is an positive number so, for a minimum distance S, S² will also be minimum

Let us S² as the function of y.

For maxima and minima,

⇒

⇒

⇒

⇒

⇒ y³ - 64 = 0

⇒

On solving we get

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Now differentiating again

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At y = 4

⇒

⇒

We get minimum distance at y = 4

Let find the value of x at these y values

⇒

⇒ x = 4

∴ The nearest point to the point (2, - 8) on the curve is (4,4).