Question

Find the points on the given curve where the tangent line is horizontal or vertical.

r = 3 cos θ

Answer 1

Step 1:

The polar equation is .

Find the points on the curve where the tangent line is horizontal or vertical.

Slope of the horizontal tangent line is 0.

Slope of the vertical tangent line is .

Find the slope of the curve.

Consider .

Differentiate on each side with respect to .

Step 2:

Slope of the tangent line is first derivative of the curve.

Slope of the parametric equation is

Substitute and in the above equation.

Slope of the tangent line is .

Comments

Contd....

Step 3:

Slope of the horizontal tangent line is 0.

The general solution of is .

If then .

If then .

Now substitute in polar equation.

Now substitute in polar equation.

The points on the curve where tangent line is horizontal are and .

---

Contd....

Step 4:

Slope of the vertical tangent line is .

The general solution of is .

If then .

If then .

Now substitute in polar equation.

Now substitute in polar equation.

The points on the curve where the tangent line is vertical are and .

Solution:

The points on the curve where the tangent line is horizontal are and .

The points on the curve where the tangent line is vertical are and .