Equation of a tangent line

Question

find the equation of the tangent line to the graph xy+y^2= 4 at x=3

Answer 1

This is not a simple equation where y is a simple function of x so it requires you to use implicit differentiation.

First, let's determine what the point on the graph is that is associated with x = 3 by plugging in 3:

(3)y + y² = 4

y² + 3y - 4 = 0

(y + 4)(y - 1) = 0

y = -4 or y = 1

So the points are (3, -4) and (3, 1) meaning there can be two answers to this problem.

To find the equation of the tangent line means you need the slope. We can get the slope from the derivative:

Plugging in the points:

y' for (3, -4) is -(-4)/(3 + 2(-4)) = 4/-5 = -4/5

and y' for (3, 1) is (-1)/(3 + 2(1)) = -1/5

So the slope is -4/5 and the point is (3, -4) so the equation of the line is y + 4 = -4/5(x -3)

And the slope is -1/5 and the point is (3, 1) so the equation of the line is y - 1 = -1/5(x -3)

Answer 2

2 tangents at x=3, y=1, y=-4

5y = 4x -7

5y = 4x -32