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

Determine the points on the curve at which the tangents to it are perpendicular to the line .

• A,
• B,
• C,
• D,

### Example

Determine the points on the curve at which the tangents to it are perpendicular to the line .

### Solution

First, find the slope function of the curve by differentiating implicitly with respect to .

Now rearrange the equation to make the subject.

Next find the slope of the given line by rearranging to make the subject.

The slope of this line is . Hence, the slope of a line perpendicular to this is .

Now set the slope function for the curve equal to this value and rearrange to form an equation connecting and .

Therefore, the - and -coordinates of the points on the curve where the tangents are perpendicular to the given line satisfy both this equation and the equation of the curve. The two equations can be solved simultaneously.

First, rearrange the linear equation to give

Now substitute this expression for into the equation of the curve and solve the resulting quadratic equation for .

Now substitute each of these -values back into the linear equation to find the -coordinate.

When :

When :

Therefore, the coordinates of the points on the curve where the tangent is perpendicular to the given line are and .

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