Question

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

Answer 1

We have y^2 = 20x .
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