# Intermediate Geometry : How to find the equation of a tangent line

## Example Questions

### Example Question #1 : How To Find The Equation Of A Tangent Line

Given the function , find the equation of the tangent line passing through .

Explanation:

Find the slope of .  The slope is 3.

Substitute  to determine the y-value.

The point is .

Use the slope-intercept formula to find the y-intercept, given the point and slope.

Substitute the point and the slope.

Substitute the y-intercept and slope back to the slope-intercept formula.

### Example Question #1 : Tangent Lines

Find the equation of the tangent line at the point  if the given function is .

Explanation:

Write  in slope-intercept form  and determine the slope.

Rearranging our equation to be in slope-intercept form we get:

.

The slope is our  value which is .

Substitute the slope and the point to the slope-intercept form.

Substitute the slope and y-intercept to find our final equation.

### Example Question #3 : How To Find The Equation Of A Tangent Line

What is the equation of the tangent line at  to the equation ?

Explanation:

Rewrite  in slope-intercept form to determine the slope. Remember slope-intercept form is .

Therefore, the equation becomes,

.

The slope is the  value of the function thus, it is .

Substitute  back to the original equation to find the value of .

Substitute the point  and the slope of the line into the slope-intercept equation.

Substitute the point and the slope back in to the slope-intercept formula.

### Example Question #4 : How To Find The Equation Of A Tangent Line

Write the equation for the tangent line of the circle through the point .

Explanation:

The circle's center is . The tangent line will be perpendicular to the line going through the points and , so it will be helpful to know the slope of this line:

Since the tangent line is perpendicular, its slope is

To write the equation in the form , we need to solve for "b," the y-intercept. We can plug in the slope for "m" and the coordinates of the point for x and y:

The equation is

### Example Question #5 : How To Find The Equation Of A Tangent Line

Find the equation for the tangent line at for the circle .

Explanation:

The circle's center is . The tangent line will be perpendicular to the line going through the points  and , so it will be helpful to know the slope of this line:

Since the tangent line is perpendicular, its slope is

To write the equation in the form , we need to solve for "b," the y-intercept. We can plug in the slope for "m" and the coordinates of the point  for x and y:

The equation is

### Example Question #6 : How To Find The Equation Of A Tangent Line

Find the equation for the tangent line of the circle  at the point .

Explanation:

The circle's center is . The tangent line will be perpendicular to the line going through the points  and , so it will be helpful to know the slope of this line:

Since the tangent line is perpendicular, its slope is .

To write the equation in the form , we need to solve for "b," the y-intercept. We can plug in the slope for "m" and the coordinates of the point  for x and y:

The equation is

### Example Question #1 : Tangent Lines

Find the equation for the tangent line to the circle at the point .

Explanation:

The circle's center is . The tangent line will be perpendicular to the line going through the points  and , so it will be helpful to know the slope of this line:

Since the tangent line is perpendicular, its slope is

To write the equation in the form , we need to solve for "b," the y-intercept. We can plug in the slope for "m" and the coordinates of the point  for x and y:

The equation is

### Example Question #8 : How To Find The Equation Of A Tangent Line

Refer to the above diagram,

Give the equation of the line tangent to the circle at the point shown.

None of the other choices gives the correct response.

Explanation:

The tangent to a circle at a given point is perpendicular to the radius that has the center and the given point as its endpoints.

The circle has its center at origin ; since this and  are the endpoints of the radius - and the line that includes this radius includes both points - its slope can be found by setting  in the following slope formula:

The tangent line, being perpendicular to this radius, has as its slope the opposite of the reciprocal of this, which is . Since the tangent line includes point , set  in the point-slope formula and simplify:

### Example Question #1 : How To Find The Equation Of A Tangent Line

A line is tangent to the circle  at the point

What is the equation of this line?

None of the other answers are correct.