# Calculus – Find the equations of both tangent lines to the graph of the ellipse that pass through the point (0,4) not on the graph.

By | November 1, 2016

# Find the equations of the tangent lines problem

In this tutorial the students will answer the question below using various techniques learned in calculus.  The students will use what they have learned about calculus, implicit differentiation and basic algebra.

Find the equations of both tangent lines to the graph of the ellipse that pass through the point (0,4) not on the graph.

$\frac { { x }^{ 2 } }{ 4 } + \frac { { y }^{ 2 } }{ 9 } = 1$

## Road map to completion

• First, begin by multiplying the entire ellipse equation by the LCD, 36.

$9{ x }^{ 2 } + 4{ y }^{ 2 } = 36$

• Second, use implicit differentiation on the ellipse equation.

$\frac { dy }{ dx } = \frac { -9x } { 4y }$

• This expression represents the slope of the tangent lines for each point of tangency around the ellipse.
• Third, find the expression for the slope of one of the tangent lines that intersects the point (0,4).

$slope = \frac { 4-y }{ -x }$

• Fourth, set those 2 expressions equal to each other and simplify.

$\frac { 4-y }{ -x } = \frac { -9x } { 4y }$
$9{ x }^{ 2 } + 4{ y }^{ 2 } = 16y$

• Simplify that equation by making a substitution using the equation from the step where we found the LCD.

$36 = 16y$

• Fifth, solve for the y variable of the point of tangency on the ellipse.

$y=\frac {9} {4}$

• Sixth, solve for the x variable of the point of tangency on the ellipse.

$x=\pm \sqrt { \frac {7} {4} }$

• Finally, use the point slope formula, with the point (0,4).

$y-4= \pm \frac {4- \frac {9} {4}}{\sqrt { \frac {7} {4} }}x$

# Graphing Tool

Click on the image below to see the results on an interactive online graphing tool

# Key Skills

## Implicit Differentiation

A special case of the chain rule used when a function cannot be written as a function of x.

## Slope Intercept Form

y=mx+b

Where m is the slope and b is the y-intercept.

## Trigonometry

Evaluating trigonometric functions

## Algebra

Factoring
Moving terms while keeping an equation balanced
Substitution
Order of Operations