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

Study concepts, example questions & explanations for Intermediate Geometry

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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 .

Possible Answers:

Correct answer:

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.

The correct answer is:  

 

Example Question #2 : How To Find The Equation Of A Tangent Line

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

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

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 #7 : How To Find The Equation Of A Tangent Line

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

Possible Answers:

Correct answer:

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

Circle

Refer to the above diagram, 

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

Possible Answers:

None of the other choices gives the correct response.

Correct answer:

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 #9 : 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?

Possible Answers:

None of the other answers are correct.

Correct answer:

Explanation:

The center of this circle is 

Therefore, the radius with endpoint  has slope  

The tangent line at  is perpendicular to this radius; therefore, its slope is the opposite of the reciprocal of , or 

Now use the point-slope formula with this slope and the point of tangency:

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