Conics, circle relationships and geometric measurement
| English | 中文 | Pinyin |
|---|---|---|
| secant/ˈsiːkənt/ | 割线 | gē xiàn |
| semiaxis/ˌsemɪˈæksɪs/ | 半轴 | bàn zhóu |
A decision before an answer
- An ellipse equation with denominators 9 and 4 has semiaxes 3 and 2, not 9 and 4.
- Your goal: Identify and interpret standard circle, ellipse, parabola and hyperbola forms.
Read the relationship
- A circle has form (x-h)²+(y-k)²=r². An ellipse uses two positive squared terms divided by squared semiaxes; a hyperbola has a difference of such terms. A parabola has one squared coordinate in its standard aligned form. In y²=4px, the vertex is the origin and the focus is (p,0); read the sign and axis before interpreting direction.
- Use circle angle, tangent and secant relationships under their conditions.
For x²/16+y²/9=1, horizontal semiaxis:
The denominator is the semiaxis squared.
Use the defining rule
- For (x-1)²/9+(y+2)²/4=1, centre is (1,-2), horizontal semiaxis 3 and vertical semiaxis 2. Complete the square when the centre is hidden in an expanded circle. An equation’s coefficients and signs distinguish conic types; a rough sketch does not supply missing algebraic information.
- Calculate area, surface area or volume with the correct dimension and units.
A cone of radius 2 and height 9 has volume:
π·2²·9/3=12π.
Check the conditions
- A tangent is perpendicular to the radius at contact. An inscribed angle is half the central angle subtending the same arc. For two secants from the same exterior point, outside segment times whole secant is equal for both; a tangent length squared equals that product. Whole length includes the outside portion, so do not multiply outside and inside alone.
- Calculate area, surface area or volume with the correct dimension and units.
The ellipse (x-1)²/9+(y+2)²/4=1 has horizontal endpoints (-2,-2) and (4,-2). y²=8x has p=2 and focus (2,0). Secants with outside/whole lengths 3/12 and 4/9 both have product 36, so a tangent from the same point has length 6. A cone radius 3, height 8 has volume 24π.
Radius of (x-2)²+(y+1)²=25 is ____.
Radius squared is 25.
Apply the task format
- Separate boundary measures from space measures. Circle circumference is 2πr and area πr²; cylinder volume is πr²h; cone volume is πr²h/3; sphere volume is 4πr³/3. Surface area sums exposed faces. If all corresponding lengths scale by k, area scales by k² and volume by k³; changing only one dimension needs the original formula.
- Calculate area, surface area or volume with the correct dimension and units.
Take square roots for radii and semiaxes; use entire secant lengths and cubic units for volume.
Which answer fits this case?
Identify and interpret standard circle, ellipse, parabola and hyperbola forms
A tangent-radius angle at contact is 90°.
They are perpendicular at that point.
Keep the distinctions
- semiaxis 半轴 — Half of an ellipse’s full principal-axis length.
- secant 割线 — A line crossing a circle at two points.
- Identify and interpret standard circle, ellipse, parabola and hyperbola forms.
- Use circle angle, tangent and secant relationships under their conditions.
- Calculate area, surface area or volume with the correct dimension and units.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.