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?

Category: QuestionsWhat is the equation of the tangent to the parabola y^2 = 8x, which is inclined at an angle of 45° with the x axis?
Editor">Editor Staff asked 11 months ago

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
 
The question was posed to me during an interview.
 
The query is from Calculus Application in chapter Application of Calculus of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

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