Question

 Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter.

Answer 1

Step 1 :  

(a)

The parametric equations are , and the point is .

Substitute the point in .

The slope of the tangent line is at .

Consider .

Apply derivative on each side with respect to t.

Consider .

Apply derivative on each side with respect to t.

  

Step 2 :

Chain rule of derivatives :

Substitute and in above expression.

Substitute in above equation.

The slope is .

The point-slope form of a line equation is .

Substitute and the point in above equation.

The tangent line equation is   

Solution :

The tangent line equation is

Answer 2

Step 1 :  

(b)

The parametric equations are , and the point is .

The slope of the tangent line is the derivative of the function at .

Consider .

Rewrite the expression :

Substitute in .

Apply derivative on each side with respect to x.

Substitute in above equation.

The slope is .

Step 2 :

The point-slope form of a line equation is .

Substitute and the point in above equation.

The tangent line equation is

Solution :  

The tangent line equation is