Two tangents to an exponential-logarithmic curve

A curve CC is given by the equation ex2y=ln⁡x2e^{x^2}y=\ln x^2, where x≠0x\ne0.

(i) Show that 2xln⁡x2+ex2dydx=2x2x\ln x^2+e^{x^2}\frac{dy}{dx}=\frac2x. marks

(ii) Find the equation of the tangent to CC at the point PP where x=1x=1. Given that the tangent at PP meets CC again at the point QQ, find the coordinates of QQ, leaving your answer correct to 5 significant figures. marks

(iii) Determine the acute angle, in degrees, between the tangents to CC at the points PP and QQ. marks

[2022/HCI/Prelim/I/9]

Total marks: 9

Solution

If this part uses an earlier result, revisit (i) Implicit differentiation with a product or open all working together.

(ii) tangent

Tangent at a point

Find the equation of the tangent to CC at the point PP where x=1x=1.

Hint

  1. Find yy at x=1x=1.

  2. Use part (i) to find the gradient there.

Answer · (ii) tangent

y=(2/e)(x−1)y=(2/e)(x-1).