# Find the equation of the tangent to the parabola, y^2 = 20 x which forms an angle 45° with the x – axis.

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Let (x1,y1) be the tangential point

Now 2yy′ = 20

∴ y′ = 10/y ie., at (x1, y1) m = 10/y1 …(1)

But the tangent makes an angle 45° with the x – axis.

∴ slope of the tangent m=tan 45° = 1 … (2)

From (1) and (2)

10/y1 = 1 ⇒ y1 = 10

But (x1,y1) lies on y^2 = 20x ⇒ y1^2 = 20 x1

100 = 20(x1) or x1 = 5

i.e., (x1,y1) = (5,10) and hence the equation of the tangent at (5, 10) is

y – 10 = 1(x – 5)

y = x + 5