What is the equation of the tangent to the parabola y^2 = 8x, which is inclined at an angle of 45° with the x axis?

(a) x + y – 2 = 0

(b) x + y + 2 = 0

(c) x – y + 2 = 0

(d) x – y – 2 = 0

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The query is from Calculus Application in chapter Application of Calculus of Mathematics – Class 12

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The correct choice is (c) x – y + 2 = 0

The best explanation: Equation of the given parabola is, y^2 = 8x ……….(1)

Differentiating both sides with respect to x,

2y(dy/dx) = 8

Or dy/dx = 4/y

Thus, equation of the tangent to the parabola (1) at (x1, y1) = (2t^2, 4t) is,

y – y1 = [dy/dx](x1, y1) (x – 2t^2)

y – 4t = [dy/dx](2(t^2), 4t) (x – 2t^2)

Putting the value of y = 4t in the equation dy/dx = 4/y, we get,

y – 4t = 4/4t(x – 2t^2) ……….(2)

If the tangent to the parabola y^2 = 8x, which is inclined at an angle of 45° with the x axis,

Then, slope of tangent (2) = tan 45° = 1

Thus, 4/4t = 1

Or t = 1

Thus, required equation of the tangent is,

y– 4 = 1(x – 2)

Putting, t = 1 in (2),

So, x – y + 2 = 0