Overview:

# Part 6: Tangents and normals

### Skill recap: Finding the equation of a line through a point with a given gradient

You should know that the line with gradient $$m$$ through the point with coordinates $$(x_1,y_1)$$ is $$y-y_1=m(x-x_1)$$. You can practise using this formula using this applet:

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### The equation of a line of the tangent or normal to a line at a given point

A tangent to a curve at a given point just touches the curve without crossing it at that point.

A normal or normal line to a curve at a given point is the line perpendicular to the tangent.

Recall that if two lines at right angles to each other have gradients $$m_1$$ and $$m_2$$, then $$m_1m_2 =-1$$ which can be rearranged as $$m_1=\dfrac{-1}{m_2}$$. In other words, the gradient of each line is the negative reciprocal of the other.

In the part 5, we saw how to calculate the gradient at a point on a curve. Using this knowledge together with the formula and facts above, we can find the equation of a tangent or a normal to a curve at a given point. In the applet below, use the toggle to change between finding a tangent and finding a normal. Drag down the slider for step-by-step solutions to each question, and generate new questions for more practice.

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