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

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(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 (x0, y0) is (x0x)/a^2 + (y0y)/b^2 = 1.

asked Jan 22, 2015 in CALCULUS by anonymous

2 Answers

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Step 1:

(a)

The ellipse is image and the point is image.

Differentiate the equation with respect to .

image

Derivative of constant is zero.

Apply formula : image.

image

Apply power rule of derivatives : image.

image

Step 2 :

Substitute the point image in the above equation.

image

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

Slope of the tangent line is image.

Step 3 :

Point-slope form of line equation is image.

Substitute image and image in the above equation.

image

The tangent line equation is image

Solution :

The tangent line equation is image

answered Jan 22, 2015 by lilly Expert
edited Jan 22, 2015 by lilly
0 votes

Step 1 :

(b)

The ellipse is image and the point is image.

Differentiate the equation with respect to .

image

Apply formula : .

Derivative of constant is zero.

image

Apply power rule of derivatives : image.

image

Step 2 :

Substitute the point image in the above equation.

image.

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

Slope of the tangent line is image.

Step 3 :

Point-slope form of line equation is image.

Substitute the point image and image in the above equation.

image

Divide each side by image.

image

The tangent line equation is image.

Solution :

The tangent line to the ellipse image at image is image.

answered Jan 22, 2015 by lilly Expert

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