Friday
Friday: The Four Conics
Friday the Boathouse wall holds four curves side by side: a circle, a parabola, the lawn oval and the clipboard hyperbola.
"Conic sections," Wren says. "All four come from slicing a cone, and all four have equations with squares."
"The circle is Geometry," Comet says. "Center and radius, from Pythagoras. We can complete the square to find them."
"The parabola has one focus and a line called the directrix," Wren says. "Equal distance to each."
Nova projects all four equations in a column. "Would you like a hint? Look at the signs on x² and y²."
"Both plus and equal: circle. Both plus, different: ellipse. One minus: hyperbola. Only one square: parabola," Comet says.
"Then let us sort the whole wall that way," Wren says.
The circle, reviewed
Take a circle with center (h, k) and radius r. A point (x, y) is on it when its distance to the center is r.
Pythagoras on the right triangle from the center gives (x - h)² + (y - k)² = r². That is the circle equation.
When the equation arrives multiplied out, complete the square. Take x² + y² - 4x + 6y - 3 = 0.
- Group: (x² - 4x) + (y² + 6y) = 3.
- Halve each middle coefficient and square it: (-4 ÷ 2)² = 4 and (6 ÷ 2)² = 9. Add both to each side.
- (x² - 4x + 4) + (y² + 6y + 9) = 3 + 4 + 9 = 16.
- Write the squares: (x - 2)² + (y + 3)² = 16.
- Center (2, -3), radius 4.
The parabola, reviewed
A parabola is the set of points equally far from a focus and a line, the directrix.
With focus (0, p) and directrix y = -p, the equation is y = x²/(4p). For p = 2, that is y = x²/8.
One square only, x², and y to the first power. That shape is how you spot a parabola.
Sorting the four
| Equation | Signs on x² and y² | Conic | The crew's version |
|---|---|---|---|
| (x - h)² + (y - k)² = r² | both plus, same divisor | circle | the turntable from Geometry |
| x²/a² + y²/b² = 1 | both plus, different divisors | ellipse | the lawn oval, a = 5, b = 3 |
| x²/a² - y²/b² = 1 | one plus, one minus | hyperbola | the clipboard, a = 3, b = 4 |
| y = x²/(4p) | only x² is squared | parabola | y = x²/8, focus (0, 2) |
Which answer makes sense?
If completing the square gives a negative number on the right, no circle exists. No real point is that far from a center.
For the oval, c = 4 must be smaller than a = 5. For the hyperbola, c = 5 must be larger than a = 3. Check both.
- Read the question.
- Tap your answer.
A dock light circle has center (2, -1) and radius 3 on the crew's grid. What is its equation?- Read the question.
- Tap your answer.
Where are the foci of the lawn oval x²/25 + y²/9 = 1?| Week review | True or false? |
|---|---|
| An ellipse is the set of points with a constant sum of distances to two foci. | ? |
| A hyperbola is the set of points with a constant difference of distances to two foci. | ? |
| For an ellipse, c² = a² + b². | ? |
| The asymptotes of x²/a² - y²/b² = 1 are y = ±(b/a)x. | ? |
| Completing the square on a circle equation finds its center and radius. | ? |
A full week of curves from two stakes. Tomorrow is Dock Day, and the family draws ovals of every shape.