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Geometry 9-12 / Week 10 / Friday
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Week 10 · Equations of Circles and Parabolas

Friday

Circles and dishes everywhere
// The turntable and the lamp dish on the stage grid
⏱ about 20 min

Friday: Circles and Dishes Everywhere

Comet spins a round tray on one finger. "Everything round in this shop has an equation now. I cannot unsee it."

"Round clock, round tray, the water ring from a cup," Wren says. "A center and a radius each."

"And the flashlight," Comet says, clicking it on. "The dish behind the bulb is a parabola. The light comes out straight."

"That is the focus doing its job," Wren says. "What do you notice about a torch beam compared with a bare bulb?"

"The bulb spreads everywhere. The dish sends it one way." She aims it at the backdrop.

Nova projects a circle and a parabola side by side. "Would you like a hint? Review time. Five ideas, every equation."

"Then next week," Wren says, "we pour water into cones. Our designer, finish the week strong."

Round and dish-shaped things

  • A round table, a clock face or the turntable: a center and a radius. The equation is (x - h)² + (y - k)² = r².
  • The ring a sprinkler wets on a lawn: every point the same distance from the sprinkler head.
  • A flashlight reflector or a satellite-style dish: every point equally far from the focus and the directrix.
  • A ball thrown gently in an arc traces a curve close to a parabola, though that is next year's story.

The week in five lines

  • A circle is a distance rule. The Pythagorean theorem turns it into (x - h)² + (y - k)² = r².
  • A point is on the circle when the sum of the squared differences equals r².
  • Complete the square, half the coefficient squared on both sides, to find a hidden center and radius.
  • A parabola is every point equally far from the focus and the directrix.
  • With the vertex at the origin and the focus at (0, p), the equation is y = x²/(4p).
EVERYDAY CIRCLES
  • Read the question.
  • Tap your answer.
A circle on a grid with center (0, 0) and radius 6A sprinkler at the origin wets a ring of radius 6 meters. Which equation describes the ring?
A round rug is drawn as x² + y² - 10x + 2y + 10 = 0. Where is its center?
The same rug is x² + y² - 10x + 2y + 10 = 0. What is its radius?
EVERYDAY DISHES
  • Read the question.
  • Tap your answer.
A parabola on a grid with its focus F at (0, 1) and the dashed directrix y = -1A small torch reflector has focus (0, 1) and directrix y = -1. What is its equation?
A dish follows y = (1/12)x² with its vertex at (0, 0). Where is its focus?
A parabola on a grid with its focus F at (0, 3) and the dashed directrix y = -3The point (6, 3) is on y = (1/12)x². How far is it from the focus (0, 3)?
Week reviewTrue or false?
The equation (x - 3)² + (y - 2)² = 25 has center (3, 2).?
The equation (x + 1)² + y² = 9 has center (1, 0).?
The point (3, 4) is on the circle with center (0, 0) and radius 5.?
To complete the square on x² + 10x, add 25.?
On a parabola, every point is closer to the focus than to the directrix.?
For y = x²/(4p), the focus is at (0, p).?
WHY THIS EXERCISEThese six checks cover everything the equations of circles and parabolas ask of you.
Which word names the fixed line of a parabola? Type the word.
WHY THIS EXERCISENaming the parts correctly is the first step of every parabola problem.
Try it
Find something round at home. Put a sheet of paper under it, mark its center and draw the circle around it.
Choose a point on the circle and measure the two legs to the center. Does the Pythagorean check give the radius?
Draw a circle and a parabola side by side. Label the center and radius on one, the focus and directrix on the other.

Week ten done, designer. Tomorrow is Shop Day: a circle in the yard and a family tape-measure check.

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