### All Intermediate Geometry Resources

## Example Questions

### Example Question #1 : Tangent Lines

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

**Possible Answers:**

**Correct answer:**

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 : Tangent Lines

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

**Possible Answers:**

**Correct answer:**

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 #1 : Tangent Lines

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

**Possible Answers:**

**Correct answer:**

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 : Tangent Lines

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

**Possible Answers:**

**Correct answer:**

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 : Tangent Lines

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

**Possible Answers:**

**Correct answer:**

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 of the circle at the point .

**Possible Answers:**

**Correct answer:**

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 : Tangent Lines

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

**Possible Answers:**

**Correct answer:**

Since the tangent line is perpendicular, its slope is

The equation is

### Example Question #8 : Tangent Lines

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:**

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 : Tangent Lines

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:**

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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