Thread: Implicit equation issue..

4. Differentiate the implicit equation of the hyperbola 4x^2 - 3y^2 = 24 to find the equation of the normal at the point (3, -2). Find the y-coordinate of the point where the normal meets the curve again.

Differentiating:

=> 8x - 6y dy/dx = 0
=> 8x = 6y dy/dx
=> dy/dx = 8x / 6y

At (3, -2), the gradient of normal is 1 / m, where gradient = 24 / -12 = -2 :
m = 1/2

So the equation of the normal is:
y + 2 = 1/2(x - 3)
y = 1/2x - 3/2 - 2
2y = x - 4

I don't know where to go from here. How to find the y-coordinate of the point where the normal meets the curve again?

Hi struck

Originally Posted by struck

y = 1/2x - 3/2 - 2
2y = x - 4

I think you want to check this once again ^^


To find the point where the normal meets the curve again, subs. the equation of the normal to the equation of the curve.

Originally Posted by songoku

Hi struck


I think you want to check this once again ^^


To find the point where the normal meets the curve again, subs. the equation of the normal to the equation of the curve.

Oh yea, it should be 2y = x - 7... But I still don't understand how to solve the next part. How do you sub. the equation of the normal to the equation of the curve

Hi struck

Equation of the curve : 4x^2 - 3y^2 = 24
Equation of normal : 2y = x - 7 ---------------> x = 2y + 7

Because the curve and the normal intersect :
4x^2 - 3y^2 = 24
4 (2y+7)^2 - 3y^2 = 24 --------> solve for y