Use implicit differentiation to find an equation of the tangent line to the ellipse x^2/2 + y^2/8 = 1 at (1, 2).

Question

(a) Use implicit differentiation to find an equation of the tangent line to the ellipse x^2/2 + y^2/8 = 1 at (1, 2).

(b) Show that the equation of the tangent line to the ellipse x^2/a^2 + y^2/b^2 = 1 at (x_0, y₀) is (x₀x)/a^2 + (y₀y)/b^2 = 1.

Answer 1

Step 1:

(a)

The ellipse is and the point is .

Differentiate the equation with respect to .

Derivative of constant is zero.

Apply formula : .

Apply power rule of derivatives : .

Step 2 :

Substitute the point in the above equation.

This is the slope of tangent to the ellipse at the point .

Slope of the tangent line is .

Step 3 :

Point-slope form of line equation is .

Substitute and in the above equation.

The tangent line equation is

Solution :

The tangent line equation is

Answer 2

Step 1 :

(b)

The ellipse is and the point is .

Differentiate the equation with respect to .

Apply formula : .

Derivative of constant is zero.

Apply power rule of derivatives : .

Step 2 :

Substitute the point in the above equation.

.

This is the slope of tangent to the ellipse at the point .

Slope of the tangent line is .

Step 3 :

Point-slope form of line equation is .

Substitute the point and in the above equation.

Divide each side by .

The tangent line equation is .

Solution :

The tangent line to the ellipse at is .