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Grade 11 Precalculus · Lesson 1.9

The Coordinate Plane & Circles

Distance and midpoint → circle standard form → writing equations → diameter problems → general form → completing the square → graph connections → tangent circles.

10 learning scenesInteractiveSame lesson material

See the right triangle

Distance Between Two Points

Interactive lab

Distance and midpoint laboratory

Move B horizontally, then calculate the squared distance AB². The graph uses equal unit lengths in both directions.

B x-coordinate = 3

\Delta x=7,\quad \Delta y=4

M=\left(\frac{-4+3}{2},\frac{1+5}{2}\right)=(-0.5,3)

Your squared distance AB²

Which midpoint coordinate stays fixed?

Explanation

The distance formula comes from the Pythagorean Theorem. Horizontal and vertical changes are the legs of a right triangle.

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Worked example · different points

Find the distance between

P(-1,-2)

and

Q(5,6)

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Coordinate changes

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Explanation

Mr. Azzam's Hint

Keep coordinate order consistent: x with x, y with y.

Your turn

For

A(1,4)

and

B(4,8)

, what is the distance?

5

7

\sqrt7

25

Average corresponding coordinates

Midpoint

Explanation

M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

The midpoint is exactly halfway between the endpoints: average x with x and y with y.

Worked example · midpoint

Find the midpoint of

C(-6,4)

and

D(2,-8)

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Average x-coordinates

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

Find the midpoint of

(2,-3)

and

(8,5)

(10,2)

(3,4)

(5,-1)

(5,1)

Equation ↔ geometry

Circle · Standard Form

Interactive lab

Circle rule laboratory

Move the center and radius. Predict which axes are tangent before checking. One square is one unit on both axes.

(x+2)^2+(y-2)^2=9

Center (-2, 2) · Radius 3

Center h = -2Center k = 2Radius r = 3

Which axes are tangent to this circle?

Explanation

A circle is the set of points the same distance from a fixed center.

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

Center $(h,k)$ · radius $r$ . The signs inside the parentheses are opposite the center coordinates.

Worked example · read a different circle

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

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Match standard form

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Explanation

Special case · center at the origin

x^2+y^2=r^2

Your turn

For

(x+5)^2+(y-1)^2=36

, find the center and radius.

(-5,1),\ r=36

(5,1),\ r=6

(-5,1),\ r=6

(5,-1),\ r=36

Build from the information given

Writing Circle Equations

Explanation

Center + radius

Identify $h,k,r$ , substitute into standard form, then simplify $r^2$ .

Worked example · center and radius

Write the equation with center

(-3,2)

and radius

5

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Start with standard form

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Explanation

Center at origin + point on circle

Use the point to find $r^2=x^2+y^2$ , then write $x^2+y^2=r^2$ .

P(-4,7)\Rightarrow r^2=16+49=65\Rightarrow x^2+y^2=65

Your turn

Write the circle with center

(1,-3)

and radius

2

(x+1)^2+(y+3)^2=4

(x-1)^2+(y+3)^2=4

(x+1)^2+(y-3)^2=4

(x-1)^2+(y-3)^2=2

Midpoint first

Diameter Problems

Explanation

Endpoints of a diameter give two jobs: midpoint → center, then distance → radius.

C=\text{midpoint},\qquad r=\frac{\text{diameter}}2

Worked example · diameter endpoints

Diameter endpoints

A(-4,1)

and

B(2,9)

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Find the center first

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Explanation

Mr. Azzam's Hint

Never use an endpoint as the center. The midpoint of the diameter is the center.

Your turn

What should you find first when diameter endpoints are given?

The midpointThe slopeThe circumferenceThe y-intercept

Reveal the hidden circle

General Form → Standard Form

Explanation

x^2+y^2+Dx+Ey+F=0

Make squared coefficients 1.
Move the constant.
Group x-terms and y-terms.
Complete both squares.
Compare with standard form.

Why are we completing the square?

We want each x-group and y-group to become a perfect-square binomial so the center and radius become visible.

The pattern

u^2+bu+\left(\frac{b}{2}\right)^2=\left(u+\frac{b}{2}\right)^2

Take half of the linear coefficient, then square it.

Step 1 · Start and move the constant

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

(x^2-4x)+(y^2+10y)=-13

Step 2 · Build the x perfect square

Show the calculation in order, then write the perfect-square binomial.

-4/2=\square

(-2)^2=\square

x^2-4x+4=(\quad)^2

Worked example · different equation

Convert

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

to standard form.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Move and group

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

After moving the constant, what should you organize before completing the square?

Expand everythingFind a midpointTake a square rootGroup x-terms and y-terms

Normalize before completing squares

When Squared Coefficients Are Not 1

Explanation

If $x^2$ and $y^2$ share the same coefficient other than 1, divide the entire equation by that coefficient first.

5x^2+5y^2-20x+10y-15=0

x^2+y^2-4x+2y-3=0

Worked example · normalized first

Find the center and radius of

2x^2+2y^2+8x-12y+6=0

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Divide every term by 2

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Explanation

Mr. Azzam's Hint

Divide every term, not only the squared terms.

Your turn

For

3x^2+3y^2+6x-y=0

, what is the first step?

Complete the square immediatelyMove only

6x

Divide every term by 3Divide only the squared terms

Move it and watch the equation

Connect the Circle Graph and Equation

8. Circle Explorer · Connect Graph and Equation

Use this exploration to connect standard form to the graph. Move the center and change the radius, and watch how the equation and graph change together. After that, preview tangency by trying to make the circle touch one or both coordinate axes.

Interactive lab

Circle explorer

Move the center and radius. Predict which axes are tangent before checking. One square is one unit on both axes.

(x+2)^2+(y-2)^2=9

Center (-2, 2) · Radius 3

Open Desmos calculator (enter the equation above)

Center h = -2Center k = 2Radius r = 3

Which axes are tangent to this circle?

Mr. Azzam's Hint

Connect the picture to the equation: the center $(h,k)$ appears with opposite signs inside the parentheses, while the right side is $r^2$ .

Your turn

If the center moves from

(0,0)

(3,-2)

while the radius stays 4, which equation matches?

x^2+y^2=16

(x-3)^2+(y+2)^2=16

(x+3)^2+(y-2)^2=16

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

Distance to an axis

Tangent Circles

Explanation

\text{tangent to y-axis}:\ r=|h|\qquad \text{tangent to x-axis}:\ r=|k|

Worked example · tangent circle

Center

(5,-2)

, tangent to the y-axis.

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Use horizontal distance

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Explanation

Mr. Azzam's Hint

y-axis → use the center's x-coordinate. x-axis → use the center's y-coordinate.

Your turn

Center

(2,-7)

, tangent to the x-axis. What is the radius?

7

2

9

5

Choose the method before calculating

Decision Lab · Exam Strategy

Center + radius

Write standard form directly

Diameter endpoints

Midpoint → center → radius

General form

Normalize → group → complete squares

Tangent to axis

Radius = perpendicular distance

Interactive lab

Circle decision laboratory

Center

(1,-3)

, radius

2

. Choose the correct equation.

Choose your prediction, then check.

Common mistake

Common mistakes

Distance: keep subtraction order consistent.
Midpoint: average corresponding coordinates.
Circle center: reverse the signs inside parentheses.
Standard form uses $r^2$ , not $r$ .
Diameter: midpoint first.
General form: normalize squared coefficients before completing squares.
Completing squares: add the same values to both sides.
Tangency: radius is perpendicular distance to the axis.

Lesson Mastery Checkpoint

Rate each skill. This is only the starting signal; your actual Practice and Assessment performance will update the plan.

Distance & midpoint

Circle standard form

Center and radius

Diameter and tangent circles

Completing the square

Normalize squared coefficients

Connect circle graph and equation

Rate all skills to prepare your personalized practice path.

Distance Between Two Points

1. Coordinate changes

2. Substitute

3. Simplify

Your turn

Your turn · student questions

Midpoint

1. Average x-coordinates

2. Average y-coordinates

3. Write the point

Your turn

Your turn · student questions

Circle · Standard Form

1. Match standard form

2. Read the center

3. Read the radius

Your turn

Your turn · student questions

Writing Circle Equations

1. Start with standard form

2. Substitute

3. Simplify

Your turn

Your turn · student questions

Diameter Problems

1. Find the center first

2. Find the diameter

3. Find the radius

4. Write the equation

Your turn

Your turn · student questions

General Form → Standard Form

1. Move and group

2. Complete both squares

3. Standard form

4. Read the circle

Your turn

Your turn · student questions

When Squared Coefficients Are Not 1

1. Divide every term by 2

2. Move and complete squares

3. Read the circle

Your turn

Your turn · student questions

Connect the Circle Graph and Equation

8. Circle Explorer · Connect Graph and Equation

Your turn

Your turn · student questions

Tangent Circles

1. Use horizontal distance

2. Write standard form

Your turn

Your turn · student questions

Decision Lab · Exam Strategy