# ACT Math : How to find the equation of a tangent line

## Example Questions

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### Example Question #181 : Lines

Circle A is centered about the origin and has a radius of 5. What is the equation of the line that is tangent to Circle A at the point (–3,4)?

–3x + 4y = 1

3x + 4y = 7

3x – 4y = –25

3x – 4y = –1

3x – 4y = –25

Explanation:

The line must be perpendicular to the radius at the point (–3,4). The slope of the radius is given by

The radius has endpoints (–3,4) and the center of the circle (0,0), so its slope is –4/3.

The slope of the tangent line must be perpendicular to the slope of the radius, so the slope of the line is ¾.

The equation of the line is y – 4 = (3/4)(x – (–3))

Rearranging gives us: 3x – 4y = -25

### Example Question #1 : Coordinate Plane

Give the equation, in slope-intercept form, of the line tangent to the circle of the equation

at the point .

None of the other responses gives the correct answer.

Explanation:

The graph of the equation  is a circle with center .

A tangent to this circle at a given point is perpendicular to the radius to that point. The radius with endpoints  and  will have slope

,

so the tangent line has the opposite of the reciprocal of this, or , as its slope.

The tangent line therefore has equation

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

Give the equation, in slope-intercept form, of the line tangent to the circle of the equation

at the point .

None of the other responses gives the correct answer.

Explanation:

Rewrite the equation of the circle in standard form to find its center:

Complete the square:

The center is

A tangent to this circle at a given point is perpendicular to the radius to that point. The radius with endpoints  and  will have slope

,

so the tangent line has the opposite of the reciprocal of this, or , as its slope.

The tangent line therefore has equation

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

What is the equation of a tangent line to

at point  ?

Explanation:

To find an equation tangent to

we need to find the first derviative of this equation with respect to  to get the slope  of the tangent line.

So,

due to power rule .

First we need to find our slope by plugging our  into the derivative equation and solving.

Thus, the slope is

.

To find the equation of a tangent line of a given point  we plug the point into

.

Therefore our equation becomes,

Once we rearrange, the equation is

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

What is the tangent line equation of

at point

?

Explanation:

To find an equation tangent to

we need to find the first derviative of this equation with respect to  to get the slope  of the tangent line.

So,

due to power rule .

First we need to find the slope by plugging our  into the derivative equation and solving.

Thus, the slope is

.

To find the equation of a tangent line of a given point  we plug it into

.

Therefore our equation is

Once we rearrange, the equation is

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

Find the equation of a tangent line to

for the point

?

Explanation:

To find an equation tangent to

we need to find the first derviative of this equation with respect to  to get the slope  of the tangent line.

So,

due to power rule .

First we need to find the slope by plugging in our  into the derivative equation and solving.

Thus, the slope is

.

To find the equation of a tangent line of a given point  we plug our point into

.

Therefore our equation is

Once we rearrange, the equation is

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

What is the equation of a tangent line to

at the point

?

Explanation:

To find an equation tangent to

we need to find the first derviative of this equation with respect to  to get the slope  of the tangent line.

So,

due to power rule .

First we need to find our slope by plugging in our  into the derivative equation and solving.

Thus, the slope is

.

To find the equation of a tangent line of a given point  we plug the point into

.

Therefore our equation is

Once we rearrange, the equation is

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

Find the equation of a tangent line to

at the point

?

Explanation:

To find an equation tangent to

we need to find the first derviative of this equation with respect to  to get the slope  of the tangent line.

So,

due to power rule .

First we need to find the slope by plugging in our  into the derivative equation and solving.

Thus, the slope is

.

To find the equation of a tangent line of a given point  we plug our point into

.

Therefore our equation is

Once we rearrange, the equation is

### Example Question #9 : How To Find The Equation Of A Tangent Line

What is the equation of a tangent line to

at point

?

Explanation:

To find an equation tangent to

we need to find the first derviative of this equation with respect to  to get the slope  of the tangent line.

So,

due to power rule .

First we need to find our slope by plugging in our  into the derivative equation and solving.

Thus, the slope is

.

To find the equation of a tangent line of a given point , we plug the point into

.

Therefore our equation is

Once we rearrange, the equation is

### Example Question #10 : How To Find The Equation Of A Tangent Line

Find the tangent line equation to

at point

?

Explanation:

To find an equation tangent to

we need to find the first derviative of this equation with respect to  to get the slope  of the tangent line.

So,

due to power rule .

First we need to find our slope at our  by plugging in the value into our derivative equation and solving.

Thus, the slope is

.

To find the equation of a tangent line of a given point

We plug into

.

Therefore our equation is

Once we rearrange, the equation is

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