Find the coordinates of a point on the parabola y = x² + 7x + 2 which is closest to the straight line y = 3x – 3.

Given equation of the parabola is y = x² + 7x + 2 and straight line is y = 3x - 3 or 3x - y - 3 = 0

Equation of parabola y = x² + 7x + 2 …… (1)

Let us assume the point on parabola which is closest to the line be (x, y)

We know that distance between the point (x₀, y₀) and the line ax + by + c = 0 is

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Now we find the distance between the line and point on parabola

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From(1)

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⇒

⇒

Let us assume S be the function of x

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For maxima and minima

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⇒

⇒ 2x + 4 = 0

⇒ 2x = - 4

⇒ x = - 2

Now differentiating again

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At x = - 2

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We get the minimum distance at x = - 2

Let’s find the value of y at this x value

⇒ y = ( - 2)² + 7( - 2) + 2

⇒ y = 4 - 14 + 2

⇒ y = - 8

∴ The point ( - 2, - 8) is the nearest point on the parabola to the line y = 3x - 3.