A tangent to the parabola y=3x^2 - 7x + 5 is perpendicular to x + 5y - 10 = 0. Determine the equation of the tangent.

Expert Answers

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The product of the slope of two perpendicular lines is -1. First find the slope of the line x + 5y - 10 = 0.

Rewrite the equation in the slope intercept form y = (-1/5)x + 2

The slope of the line perpendicular to this is 5.

The slope of the tangent to the parabola at any point x, is given by the value of the derivative of y.

y' = 6x - 7

6x - 7 = 5

=> x = 12

=> x = 2

For x = 2, y = 3*2^2 - 7*2 + 5 = 12 - 14 + 5 = 3

The equation of the line passing through (2, 3) which has a slope 5 is (y - 3)/(x - 2) = 5

=> y - 3 = 5x - 10

=> y - 5x + 7 = 0

The equation of the required tangent is y - 5x + 7 = 0

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