Coordinate geometry and tangents
Введено| English | Значение |
|---|---|
| gradient/ˈɡreɪdɪənt/ | угловой коэффициент |
How does a path's slope become an equation?
- A path rises 6 metres over a horizontal distance of 3 metres. Its gradient connects a diagram to an equation.
- This lesson studies gradient 斜率: The change in y divided by the corresponding change in x.
The method
- A line through (x₁,y₁) with gradient m has y-y₁=m(x-x₁).
- Parallel lines have equal gradients.
- Finite perpendicular gradients multiply to -1.
- A circle has (x-a)²+(y-b)²=r².
Two non-vertical lines are perpendicular. What is true of their gradients?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Perpendicular gradients satisfy m₁m₂=-1: each is the negative reciprocal of the other. Equal gradients give parallel lines instead.
A worked case
Through (2,5) with gradient 3, y-5=3(x-2), so y=3x-1. A perpendicular through the same point has y-5=-(x-2)/3. The circle (x-2)²+(y+1)²=25 has centre (2,-1) and radius 5.
Coordinate geometry and tangents
A line through (x₁,y₁) with gradient m has y-y₁=m(x-x₁)
Compare the model with the worked case and explain one change.
Find the gradient through (1,2) and (4,8).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Gradient=(8-2)/(4-1)=6/3=2.
Where it goes wrong
- A vertical line has no finite gradient; do not force it into y=mx+c.
- Read the signs of a circle's centre carefully.
- The radius to a tangent is perpendicular to the tangent.
Not true: Perpendicular nonvertical lines have equal gradients. Perpendicular gradients multiply to −1: a gradient of 2 needs −1/2. Parallel lines have equal gradients.
Find the y-intercept of y-5=3(x-2).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Expand y-5=3x-6, so y=3x-1; the intercept is -1.
Perpendicular nonvertical lines have equal gradients.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Perpendicular gradients multiply to −1: a gradient of 2 needs −1/2. Parallel lines have equal gradients.
Applying it
- Before solving a line-circle intersection, predict whether there are zero, one or two intersections.
- Substitution produces a quadratic whose discriminant checks the prediction.
Find the radius of (x-2)²+(y+1)²=25.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
The radius is the positive square root of 25, namely 5.
Match each feature of y=mx+c to its meaning.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Gradient measures steepness and the intercept fixes position; equal gradients with different intercepts never meet.
In your course
- AQA A-level Mathematics (7357), specification section C.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Gradient: The change in y divided by the corresponding change in x.
- A line through (x₁,y₁) with gradient m has y-y₁=m(x-x₁).
- Watch for: A vertical line has no finite gradient; do not force it into y=mx+c.