Calculus problem?

Question

a) A point moves on the hyperbola 3x^2 - y^2 = 23 so that its y-coordinate is increasing at a constant rate of 4 units per second. How fast is the x-coordinate changing when x = 4?
b) For what values of k will the line 2x + 9y + k = 0 be normal to the hyperbola 3x^2 - y^2 = 23?

Answer 1

(a)
3x^2 - y^2 = 23 ...(1)
6x - 2y dy/dx = 0
dy/dx = 3x / y

When x = 4, (1) gives:
y^2 = 3*4^2 - 23
= 25
y = +/- 5.

(dy/dt) / (dx/dt) = 3x / y

When dy/dt = 4 units/sec. and x = 4:
4 / (dx/dt) = 3 * 4 / (+/- 5)
dx / dt = 4 * (+/- 5) / (3 * 4)
= +/- 5 / 3.

(b)
The gradient of a normal is:
- 1 / (dy/dx)
= - y / 3x

2x + 9y + k = 0 ...(2)
9y = - 2x - k
y = -(2/9)x - k/9
The gradient of this line is - 2 / 9.

It is a normal to the hyperbola when:
y / 3x = 2 / 9
9y = 6x
y = 2x / 3 ...(3)

Substituting for y in (1):
3x^2 - 4x^2 / 9 = 23
27x^2 - 4x^2 = 207
x = +/- 3

Substituting in (3) gives the points (3, 2) and (-3, -2),
and (2) then gives:
k = +/- 24.