ACT Circles and Conic Sections: Equations, Graphs, and Geometry

ACT Geometry currently accounts for roughly 17–20% of Math, and the official content description includes circles, missing values in geometric figures, trigonometric ratios, and equations of conic sections.

For circles, the highest-value equation is:

$$\boxed{(x-h)^2+(y-k)^2=r^2}$$

where:

• center = $(h,k)$;
• radius = $r$.

The most common ACT trap is the sign inside the parentheses.

Reading a circle equation

Example:

$$(x-3)^2+(y+2)^2=25$$

Center:

$$\boxed{(3,-2)}$$

Radius:

$$\boxed{5}$$

Why center y-coordinate is $-2$:

$$y+2=y-(-2)$$

Inside signs reverse.

Writing a circle equation from center and radius

Center:

$$(-4,1)$$

radius:

$$3$$

Equation:

$$\boxed{(x+4)^2+(y-1)^2=9}$$

Do not forget:

right side is $r^2$, not $r$.

Completing the square to find center

Suppose:

$$x^2+y^2-6x+8y-11=0$$

Group:

$$x^2-6x+y^2+8y=11$$

Complete each square:

$$x^2-6x+9=(x-3)^2$$ $$y^2+8y+16=(y+4)^2$$

Add 9 and 16 to both sides:

$$(x-3)^2+(y+4)^2=36$$

So:

$$\boxed{\text{center }(3,-4),\quad r=6}$$

This is a classic ACT algebra-geometry bridge.

Point on a circle

To test whether a point lies on a circle, substitute.

Circle:

$$(x-2)^2+(y+1)^2=25$$

Point:

$$(5,3)$$

Compute:

$$(5-2)^2+(3+1)^2 = 9+16 = 25$$

So the point lies on the circle.

Distance from center gives radius

If center is:

$$(1,-2)$$

and point on circle is:

$$(5,1)$$

radius:

$$r=\sqrt{(5-1)^2+(1+2)^2}$$ $$=\sqrt{16+9}=5$$

Then equation:

$$(x-1)^2+(y+2)^2=25$$

Tangent and radius are perpendicular

A tangent line touches a circle at one point.

The radius to that point is perpendicular to the tangent.

If radius slope is:

$$m$$

tangent slope is:

$$-\frac1m$$

when both slopes are defined and nonzero.

This combines coordinate geometry with circle properties.

Arc length

For central angle $\theta$ in degrees:

$$\boxed{\text{arc length}=\frac{\theta}{360^\circ}(2\pi r)}$$

Example:

radius $=12$, angle $=60^\circ$:

$$\frac{60}{360}(24\pi) = \boxed{4\pi}$$

Sector area

$$\boxed{\text{sector area}=\frac{\theta}{360^\circ}(\pi r^2)}$$

Same fraction of the full circle.

For $r=6$, $\theta=90^\circ$:

$$\frac14(36\pi)=\boxed{9\pi}$$

Central and inscribed angles

A central angle has vertex at the center.

An inscribed angle has vertex on the circle.

For the same intercepted arc:

$$\boxed{\text{inscribed angle}=\frac12(\text{central angle})}$$

If central angle is $100^\circ$:

$$\text{inscribed}=50^\circ$$

Conic sections: what ACT expects

The four classical conics are:

• circle;
• parabola;
• ellipse;
• hyperbola.

You do not need a college analytic-geometry course.

You should recognize standard forms and graph features.

Parabola

Common vertical form:

$$y=a(x-h)^2+k$$

Vertex:

$$(h,k)$$

If:

• $a>0$: opens up;
• $a<0$: opens down.

This overlaps ACT Functions and Quadratics.

Horizontal parabola

$$x=a(y-k)^2+h$$

opens left/right depending on sign of $a$.

If ACT asks only graph recognition, identify:

• vertex;
• orientation;
• symmetry.

Ellipse standard form

$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} =1$$

Center:

$$(h,k)$$

The larger denominator corresponds to the longer semi-axis.

Example:

$$\frac{(x-2)^2}{25}+\frac{(y+1)^2}{9}=1$$

Center:

$$(2,-1)$$

Horizontal semi-major axis:

$$5$$

Vertical semi-minor axis:

$$3$$

Hyperbola recognition

Horizontal-opening form:

$$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1$$

Vertical-opening form:

$$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} =1$$

The positive term tells the opening direction.

You may be asked to identify which graph matches an equation.

Circle vs ellipse

Circle:

$$\frac{(x-h)^2}{r^2} + \frac{(y-k)^2}{r^2} =1$$

Both denominators equal.

Ellipse:

denominators differ.

A circle is a special ellipse.

Completing the square for conics

You may be given a nonstandard equation and asked for graph features.

Strategy:

1. group x-terms;
2. group y-terms;
3. complete squares;
4. move constant;
5. divide to get right side 1 if useful.

Do not memorize a separate trick for every conic.

Use algebra.

Worked example: circle from expanded form

$$x^2+y^2+4x-10y+13=0$$

Move constant:

$$x^2+4x+y^2-10y=-13$$

Complete:

$$(x+2)^2+(y-5)^2=16$$

Center:

$$\boxed{(-2,5)}$$

Radius:

$$\boxed{4}$$

Worked example: ellipse features

$$\frac{x^2}{16}+\frac{y^2}{4}=1$$

Center:

$$(0,0)$$

Horizontal endpoints:

$$(\pm4,0)$$

Vertical endpoints:

$$(0,\pm2)$$

Worked example: parabola shift

$$y=-2(x+3)^2+5$$

Vertex:

$$\boxed{(-3,5)}$$

Opens:

downward.

Vertical stretch:

factor 2.

Common ACT circle traps

Trap 1

Center signs not reversed.

Trap 2

Using radius instead of $r^2$ on right side.

Trap 3

Forgetting to add completing-square constants to both sides.

Trap 4

Confusing diameter with radius.

Trap 5

Using arc formula for sector area or vice versa.

Common conic traps

Trap 1

Calling every squared equation a circle.

Trap 2

Ignoring sign between squared terms.

Trap 3

Using smaller denominator as major axis.

Trap 4

Missing vertex/center shifts.

Trap 5

Trying to memorize graph shape without rewriting equation.

Fast ACT circle routine

If equation is standard:

1. read center;
2. square-root right side for radius.

If expanded:

1. complete square;
2. rewrite standard form;
3. read features.

If geometry:

1. identify radius/diameter;
2. decide whether arc, sector, tangent, or angle relationship is being tested.

Fast conic-recognition routine

Circle

two squared terms, same coefficient/sign after standardization.

Ellipse

two positive squared terms, different scales.

Hyperbola

squared terms subtract.

Parabola

only one variable squared in the simplest 2D form.

Current ACT relevance

Current ACT Math explicitly includes:

• circles;
• missing values in triangles, circles, and other figures;
• trigonometric ratios;
• equations of conic sections.

Geometry is about 17–20% of current Math content.

Practice set

1

$$(x+1)^2+(y-4)^2=49$$

Center:

$$\boxed{(-1,4)}$$

Radius:

$$\boxed{7}$$

2

A $120^\circ$ arc in a circle of radius 9:

$$\frac{120}{360}(18\pi)=\boxed{6\pi}$$

3

$$\frac{x^2}{36}+\frac{y^2}{9}=1$$

Major semi-axis:

$$\boxed{6}$$

horizontal.

How to review mistakes

Bad note:

circle equation error.

Better:

I read $y+4$ as center $+4$ instead of $-4$.

Or:

I used the circumference fraction for sector area without multiplying by $\pi r^2$.

Coordinate geometry with circles

Suppose a circle has diameter endpoints:

$$(-2,4)\quad\text{and}\quad(6,-2)$$

Center is midpoint:

$$\left(\frac{-2+6}{2},\frac{4+(-2)}{2}\right) = (2,1)$$

Diameter length:

$$\sqrt{(8)^2+(-6)^2} = 10$$

Radius:

$$5$$

Equation:

$$\boxed{(x-2)^2+(y-1)^2=25}$$

This combines midpoint, distance, and circle equation in one problem.

Tangent-line slope example

Circle center:

$$(1,2)$$

point of tangency:

$$(5,4)$$

Radius slope:

$$\frac{4-2}{5-1}=\frac12$$

Tangent slope:

$$\boxed{-2}$$

ACT often rewards chaining two familiar rules rather than using a special tangent formula.

Chords and perpendicular bisectors

A radius drawn perpendicular to a chord bisects the chord.

If a chord has length 12 and perpendicular distance from center to chord is 5, half-chord is 6.

Radius forms a right triangle:

$$r^2=5^2+6^2$$ $$r=\sqrt{61}$$

This is a common geometry blend.

Equation classification before solving

Example:

$$4x^2+4y^2-16x+8y=20$$

Same coefficients on $x^2$ and $y^2$, same sign.

That strongly suggests:

circle.

Divide by 4:

$$x^2+y^2-4x+2y=5$$

Then complete squares.

Classification first helps choose the right algebra.

Ellipse endpoints

For:

$$\frac{(x-1)^2}{16} + \frac{(y+2)^2}{25} =1$$

Center:

$$(1,-2)$$

Larger denominator 25 is under y-term.

So major axis is vertical.

Vertices:

$$(1,-2\pm5)$$ $$\boxed{(1,3),(1,-7)}$$

Hyperbola orientation

$$\frac{(y-4)^2}{9} - \frac{(x+2)^2}{16} =1$$

Positive term is y-term.

So hyperbola opens:

up and down.

Center:

$$(-2,4)$$

Do not let the larger denominator determine orientation. The sign determines it.

Parabola and conic overlap with functions

For:

$$y=(x-3)^2-7$$

you can treat it as:

• a conic (parabola);
• a transformed quadratic function.

The best method depends on the question.

If asked vertex:

read transformation.

If asked intersections:

solve system.

If asked graph:

use orientation and vertex.

Circle inequality

A related harder concept:

$$(x-h)^2+(y-k)^2<r^2$$

represents points inside the circle.

$$>r^2$$

represents points outside.

This can appear in systems/regions questions.

Review checklist

Classify misses as:

• sign reversal in center;
• radius vs radius-squared;
• complete-square error;
• wrong conic classification;
• wrong major-axis direction;
• tangent/perpendicular error;
• arc/sector confusion.

That makes review much more precise.

Circle intersections with lines

ACT can combine circle equations with a line.

Example:

$$x^2+y^2=25$$

and:

$$y=4$$

Substitute:

$$x^2+16=25$$ $$x^2=9$$ $$x=\pm3$$

Intersections:

$$\boxed{(3,4),(-3,4)}$$

This is a system-of-equations problem disguised as geometry.

Tangency as one intersection

A line tangent to a circle intersects it at exactly one point.

If substitution into the circle creates a quadratic with discriminant:

$$D=0,$$

the line is tangent.

That connects:

• circle geometry;
• systems;
• discriminant.

Circle scaling and area

If radius doubles:

$$r\rightarrow2r$$

circumference doubles:

$$2\pi r\rightarrow4\pi r$$

but area becomes four times as large:

$$\pi r^2\rightarrow4\pi r^2.$$

ACT often tests geometric scaling rather than direct formula substitution.

Ellipse area

For ellipse with semi-axes $a$ and $b$:

$$\boxed{A=\pi ab}$$

Example:

$$\frac{x^2}{25}+\frac{y^2}{9}=1$$

semi-axes:

$$5,\ 3$$

area:

$$\boxed{15\pi}$$

This formula is useful when the equation is already in standard form.

Conic equation checklist

Before doing algebra, inspect:

• Are both $x^2$ and $y^2$ present?
• Same sign?
• Same coefficient?
• Opposite signs?
• Only one squared variable?

That lets you classify the conic before completing the square.

Same sign, same coefficient

Likely circle.

Same sign, different coefficients

Likely ellipse.

Opposite signs

Hyperbola.

One squared variable

Parabola.

Hyperbola asymptote awareness

For:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

asymptotes are:

$$y=\pm\frac{b}{a}x$$

when centered at origin.

You do not need to over-memorize shifted formulas, but knowing hyperbolas approach asymptotes helps identify graphs quickly.

Conic review framework

If you miss a conic question, record whether the error was:

• classification;
• completing square;
• sign of center;
• radius/semi-axis;
• orientation;
• tangent/intersection;
• scaling.

That makes later practice much more targeted.

One more expert shortcut: count intersections before solving

When a line and a conic are shown graphically, first ask how many intersections are even possible.

For a circle and line:

• 0 intersections → line misses circle;
• 1 → tangent;
• 2 → secant.

For a parabola and line:

• 0, 1, or 2 real intersections.

For two graphs described by equations, substitution often produces a quadratic. Its discriminant tells the same story:

$$D>0 \rightarrow 2\text{ real intersections}$$ $$D=0 \rightarrow 1\text{ tangent intersection}$$ $$D<0 \rightarrow 0\text{ real intersections}$$

This lets you answer some geometry questions without computing every coordinate.

FAQ: ACT circles and conics

Are conic sections still on ACT Math?

Yes. Current ACT Math explicitly includes equations of conic sections.

What circle formula should I know?

$$(x-h)^2+(y-k)^2=r^2$$

How do I distinguish ellipse and hyperbola?

Ellipse has two added squared terms; hyperbola has a subtraction between squared terms in standard form.

Does ACT test completing the square?

It can be useful for converting circle/conic equations to standard form and for quadratics.

Related MastaPrep guides

• ACT Geometry and Trigonometry
• ACT Quadratics
• ACT Function Composition and Transformations
• ACT Trigonometry
• Enhanced ACT 2026–27 Guide

Final takeaway

For circles:

$$\boxed{(x-h)^2+(y-k)^2=r^2}$$

For conics, classify by squared terms and signs before doing heavy algebra.

Most ACT difficulty comes from:

• sign reading;
• completing the square;
• identifying graph features;
• choosing the right circle relationship.

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Official ACT references:
https://www.act.org/content/act/en/products-and-services/the-act/test-preparation/description-of-math-test.html
https://www.act.org/content/dam/act/unsecured/documents/R2519-Design-Framework-for-the-ACT-Enhancements-2026-02.pdf

Trademark note: ACT® is a registered trademark of ACT Education Corp. MastaPrep is independent and is not affiliated with or endorsed by ACT Education Corp.