- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 4
Differentiation rules practice question 20
The normal line to the curve at the point is the line through perpendicular to the tangent line there. It meets the curve at a second point. Find the -coordinate of that point.
The normal is perpendicular to the tangent at . How are the slopes of perpendicular lines related, and where does the tangent slope come from?
, so the tangent slope at is . The normal slope is its negative reciprocal.
Set the normal line equal to the curve. One solution is , which you already know, so the quadratic has as a factor.
An exact value, or a decimal.
Theory UBC Math 100 §4.1CLP-1 §2.4Key results
Worked solution
Strategy
Find the normal line at (2, 2), then solve for where it meets the parabola again. Knowing one intersection, x = 2, makes the quadratic easy to factor.
Working
The normal slope is the negative reciprocal of the tangent slope.
Point-slope form through (2, 2).
Set the curve equal to the line, multiply by 5 and collect.
x = 2 is the point we started from; the other root is the new point.
Answer
The answer is negative three fifths.
Check
At x = −0.6 the curve and the normal line both give y = 2.52.
Where this goes wrong
Using the tangent line instead of the normal. The tangent meets this parabola only at x = 2, a double root, so there is no second point to find.
Leans on
Perpendicular slopes, from Functions, graphs and inverses
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