The normal at a point is perpendicular to the tangent there. If the tangent's gradient is m, the normal's gradient is −1/m.
For y = 4x² − x + 4 at x = 1: y = 7 and dy/dx = 8x − 1 = 7, so the normal's gradient is −1/7. The normal is y − 7 = −(1/7)(x − 1), which rearranges to x + 7y − 50 = 0. Questions often ask for the form ax + by + c = 0 with whole numbers.
Worked example
The gradient of a curve at (2, 5) is 3. Find the equation of the normal.
- Normal gradient: −1/3.
- y − 5 = −(1/3)(x − 2), so 3y − 15 = −x + 2.
Answer:x + 3y − 17 = 0
Key words
- normal
- the line perpendicular to the tangent
- negative reciprocal
- −1/m
- ax + by + c = 0
- a common form for line equations
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