Coordinate geometry and tangents
แนะนำใหม่| English | ความหมาย |
|---|---|
| gradient/ˈɡreɪdɪənt/ | ความชัน |
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?
ความยาก 1/5 · 100 เหรียญ + โบนัสเล็กน้อยในการทำครั้งแรกวันนี้
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).
ความยาก 2/5 · 140 เหรียญ + โบนัสเล็กน้อยในการทำครั้งแรกวันนี้
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).
ความยาก 2/5 · 140 เหรียญ + โบนัสเล็กน้อยในการทำครั้งแรกวันนี้
Expand y-5=3x-6, so y=3x-1; the intercept is -1.
Perpendicular nonvertical lines have equal gradients.
ความยาก 1/5 · 100 เหรียญ + โบนัสเล็กน้อยในการทำครั้งแรกวันนี้
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.
ความยาก 2/5 · 140 เหรียญ + โบนัสเล็กน้อยในการทำครั้งแรกวันนี้
The radius is the positive square root of 25, namely 5.
Match each feature of y=mx+c to its meaning.
ความยาก 2/5 · 140 เหรียญ + โบนัสเล็กน้อยในการทำครั้งแรกวันนี้
Gradient measures steepness and the intercept fixes position; equal gradients with different intercepts never meet.
In your course
- Pearson Edexcel International A Level Pure Mathematics, unit P2.
- 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.
Write from a radius of 6
- The circle is (x minus 1) squared plus (y plus 2) squared equals 36.
- Find the radius.
What is the radius of (x minus 1) squared plus (y plus 2) squared equals 36?
The square root of 36 is 6.
What is the radius?
ความยาก 1/5 · 100 เหรียญ + โบนัสเล็กน้อยในการทำครั้งแรกวันนี้
Radius squared is 36. Other options — 5 (the lesson radius); 36 (the right-hand side).