Origin-centred circles and tangent equations · Higher
Diperkenalkan| English | Makna |
|---|---|
| radius gradient/ˈreɪdɪəs ˈɡreɪdɪənt/ | gradien jari-jari |
Which line just touches the circular path?
- A circular path has a sensor at (3,4). Which straight line just touches the path at that point?
- This lesson studies radius gradient 半径斜率: The slope of a radius joining the centre to a point on the circle.
The method
- For an origin-centred circle, x²+y²=r².
- A point is on the circle if its squared coordinates sum to r².
- A tangent is perpendicular to the radius there.
- Use a negative reciprocal gradient when both gradients are finite; handle horizontal/vertical cases directly.
Why is the tangent gradient at (3,4) on x²+y²=25 equal to -3/4?
Kesulitan 1/5 · 100 koin + bonus kecil pada penyelesaian pertama hari ini
The radius from (0,0) to (3,4) has gradient 4/3. The radius-tangent theorem forces perpendicularity, so the tangent gradient is its negative reciprocal, -3/4.
A worked case
At (3,4), r²=3²+4²=25, so x²+y²=25. The radius gradient is 4/3, hence tangent gradient -3/4. Its equation y-4=(-3/4)(x-3) simplifies to 3x+4y=25. At (5,0) the radius is horizontal and the tangent is the vertical line x=5.
Origin-centred circles and tangent equations
For an origin-centred circle, x²+y²=r²
Compare the model with the worked case and explain one change.
Find the radius of the circle through (3,4).
Kesulitan 2/5 · 140 koin + bonus kecil pada penyelesaian pertama hari ini
√(3²+4²)=5.
Where it goes wrong
- The radius and tangent share a point but different directions.
- Do not use the negative reciprocal of zero; a horizontal radius has a vertical tangent.
- Check the tangent passes through its contact point.
Not true: The tangent at a circle point has the same gradient as the radius to that point. The tangent is perpendicular to the radius, so its gradient is the negative reciprocal of the radius gradient.
Find the tangent gradient at (3,4).
Kesulitan 2/5 · 140 koin + bonus kecil pada penyelesaian pertama hari ini
The negative reciprocal of 4/3 is -3/4.
The tangent is perpendicular to the radius, so its gradient is the negative reciprocal of the radius gradient.
Kesulitan 1/5 · 100 koin + bonus kecil pada penyelesaian pertama hari ini
True. The tangent is perpendicular to the radius, so its gradient is the negative reciprocal of the radius gradient.
Applying it
- AQA A16 Higher concerns circles centred at the origin.
- Offset-centre circle formulae and circle calculus are not needed for this lesson.
For tangent 3x+4y=25, find x when y=0. Give it to 3 significant figures.
Kesulitan 2/5 · 140 koin + bonus kecil pada penyelesaian pertama hari ini
Set y=0, then divide 25 by 3.
Match each circle fact for x²+y²=r² to its statement.
Kesulitan 2/5 · 140 koin + bonus kecil pada penyelesaian pertama hari ini
The negative-reciprocal rule covers finite nonzero gradients; horizontal or vertical cases are handled directly.
In your course
- AQA GCSE Mathematics (8300), Higher tier, specification section 3.2.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Radius gradient: The slope of a radius joining the centre to a point on the circle.
- For an origin-centred circle, x²+y²=r².
- Watch for: the radius and tangent share a point but different directions.