Question

For the curve y = 4x³ – 2x⁵, find all the points at which the tangent passes through the origin.

Answer 1

y = 4x³ – 2x⁵ dy 

\(\frac{dy}{dx}\)= 12x² – 10x⁴dx 

Let (a, b) be the point on the curve at which the tangent passes through the origin. 

∴ Equation of tangent is 

y – b = (12a² – 10a²) (x – a) 

but this passes through the origin 

∴ 0 – b = (12a² – 10a⁴) (-a) 

b = 12a³ – 10a⁵ ….(1) 

Also from the equation b = 4a³ – 2a⁵ …. (2) 

from (1) and (2) 12a³ – 10a⁵ = 4a3 – 2a⁵ 8a³ = 8a⁵ 

a^{3 (}1 – a²) = 0 ⇒ a = 0, a = + 1 

when a = 0, b = 0, (0, 0) 

a = 1,b= 12(1)- 10(1) = 2, (1,2) 

a = -1, b = 12 (-1) -10 (-1) = -2, (-1, -2) 

Hence the required points are (0,0), (1,2), (-1,-2).