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.