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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.5, Problem 52E

To determine

**To find:** The equation of the both tangent lines to the ellipse passing through the point.

Expert Solution

The equation of the tangent line to the curve passing through the point

**Given:**

The equation of the ellipse

The point is

**Derivative rules:**

*Chain rule*:

**Calculation:**

Obtain the slope of tangent to the equation.

Differentiate *x*.

Apply the chain rule and simplify the terms.

Thus, the derivative of the equation of the ellipse is

That is, the slope of the tangent is

Let

The slope of tangent to the curve at

The equation of the tangent line passing through the point

Here, the tangent line is also passing through the point

Substitute

The value of the curve

Substitute the equation (3) in equation (2).

Substitute

Simplify the quadratic equation,

Substitute

Thus, the value of *b* is 3.

Substitute

Thus, the points are

Substitute

Thus, the equation of the tangent line to the curve passing through the point

Substitute

Thus, the equation of the tangent line to the curve passing through the point