# 12.6.4. Equation of the Tangent and the Normal to a Curve

Find the equations of the tangents to the curve that are parallel to the straight line .

• A,
• B,
• C,
• D,

### Example

Find the equations of the tangents to the curve that are parallel to the straight line .

### Solution

First, find the slope of the given straight line by rearranging to make the subject.

Therefore, the slope of this straight line is 1.

Now find the slope function of the curve by differentiating with respect to .

If the tangents are parallel to the given straight line, then their slope will also be equal to 1. Set the slope function equal to 1 and solve for .

Now find the -coordinate on the curve when takes each of these values.

When :

When :

Next find the equation of each tangent.

The first tangent has a slope of 1 and passes through the point . Its equation can be found using the general equation of a straight line.

The second tangent has a slope of 1 and passes through the point . Its equation can be found using the general equation of a straight line.

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