Thirty-Six

Unit 6 · GeometryG5 · Coordinate Geometry · Aerodynamics & Navigation

Circle Equations & Conics

Read centers, radii, vertices, and axes from circle, ellipse, and hyperbola forms.

  • 30 min
  • Unlocks: Orbital Plotter
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

standard form
an equation arranged to show a conic center and scale

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Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

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The big idea

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The decision that matters

For ellipses, the larger denominator marks the major-axis direction. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

For ellipses, the larger denominator marks the major-axis direction. Keep exact values through the setup; round only when the stem asks.
Conic standard forms
circle: (x−h)2+(y−k)2=r2ellipse: (x−h)2a2+(y−k)2b2=1horizontal hyperbola: (x−h)2a2−(y−k)2b2=1\begin{aligned} \text{circle: }&(x-h)^2+(y-k)^2=r^2\\ \text{ellipse: }&\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\\ \text{horizontal hyperbola: }&\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \end{aligned}

The center is (h,k)(h,k). For an ellipse, the larger denominator gives the major-axis direction. For a hyperbola, the positive term gives the opening direction.

For example, (x−1)216−(y+2)29=1\frac{(x-1)^2}{16}-\frac{(y+2)^2}{9}=1 is centered at (1,−2)(1,-2) and opens left and right because the xx-term is positive. Since a=4a=4, its vertices are (1±4,−2)(1\pm4,-2), or (−3,−2)(-3,-2) and (5,−2)(5,-2).

Algebra or Desmos?

Standard form gives exact centers, radii, and semiaxis lengths more reliably than visual estimation. Graph the conic to confirm orientation and vertices after reading the signs and denominators algebraically.

Explore in Desmos

Graph a circle and an ellipse from standard form

Graph the conics, then read center and vertices from standard form rather than estimating pixels.

The calculator is live: change anything and see what happens.

Graph a circle and an ellipse from standard form

Degree mode

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Trap

The main trap

The signs inside parentheses oppose the center coordinates.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Find the center and radius of (x−3)2+(y+4)2=49(x-3)^2+(y+4)^2=49.
Center is (3,−4)(3,-4) because signs oppose.
r2=49r^2=49, so r=7r=7.

Which ordered pair is the center?

The center is (3,−4)(3,-4) and the radius is 7.

Why is 49 the square of the radius?

Worked example

Example 2: ACT twist

Problem

Find the center, major-axis direction, and vertices of (x+2)236+(y−1)29=1\frac{(x+2)^2}{36}+\frac{(y-1)^2}{9}=1.
Center is (−2,1)(-2,1).

Which denominator determines the major-axis direction?

The vertices are (−8,1)(-8,1) and (4,1)(4,1).

Why is the major axis horizontal?

Faded example · you finish it

Example 3: You finish it

Problem

Find the radius of x2+y2−6x+8y=0x^2+y^2-6x+8y=0.
Complete squares: (x−3)2+(y+4)2=25(x-3)^2+(y+4)^2=25.

Take the positive square root of r2=25r^2=25.

Desmos way

Verify conic centers and vertices

Graph the conics, then read center and vertices from standard form rather than estimating pixels.

Verify conic centers and vertices

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.