Measurement and Geometry: Lines and Circles - Master Reviewer

Equations of Lines

A line in a coordinate plane can be represented in various mathematical forms depending on the information available.

• Slope-Intercept Form:

  • Formula: $y = mx + b$

  • Where:

    • $m$ = slope

    • $b$ = y-intercept

  • Example: Given a slope of $3$ and a y-intercept of $-2$.

    • Substitute into formula: $y = 3x - 2$

    • Answer: $y = 3x - 2$

• Slope Formula:

  • Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

  • This formula is used specifically when the coordinates of two points on the line are known.

  • Example: Find the slope of the line passing through $(2,3)$ and $(6,11)$.

    • Calculation: $m = \frac{11 - 3}{6 - 2}$

    • Calculation: $m = \frac{8}{4}$

    • Answer: $m = 2$

• Point-Slope Form:

  • Formula: $y - y_1 = m(x - x_1)$

  • Used when one point is known and the slope is known.

  • Example: Point $(3,4)$ and Slope $5$.

    • Substitute: $y - 4 = 5(x - 3)$

    • Distribute: $y - 4 = 5x - 15$

    • Simplify: $y = 5x - 11$

    • Answer: $y = 5x - 11$

• Equation of a Line Through Two Points:

  • Example: Points $(1,2)$ and $(5,10)$.

  • Step 1: Find slope.

    • $m = \frac{10 - 2}{5 - 1}$

    • $m = \frac{8}{4} = 2$

  • Step 2: Use point-slope form.

    • $y - 2 = 2(x - 1)$

  • Step 3: Simplify.

    • $y - 2 = 2x - 2$

    • $y = 2x$

    • Answer: $y = 2x$

Parallel and Perpendicular Lines

The relationship between two lines can be determined by comparing their slopes.

• Parallel Lines:

  • Condition: $m_1 = m_2$

  • Parallel lines never intersect and have identical slopes.

  • Example: Given the line $y = 2x + 3$, find a parallel line that passes through $(4,1)$.

    • Slope remains $2$.

    • Equation: $y - 1 = 2(x - 4)$

    • Simplify: $y - 1 = 2x - 8$

    • Answer: $y = 2x - 7$

• Perpendicular Lines:

  • Condition: $m_1 \times m_2 = -1$

  • Slopes of perpendicular lines are negative reciprocals of each other.

  • Example: Given the line $y = 4x - 2$.

    • Original slope ($m_1$) = $4$

    • Perpendicular slope ($m_2$) = $-\frac{1}{4}$

    • Set through point $(2,3)$: $y - 3 = -\frac{1}{4}(x - 2)$

    • Distribute: $y - 3 = -\frac{1}{4}x + \frac{1}{2}$

    • Final calculation: $y = -\frac{1}{4}x + \frac{7}{2}$

    • Answer: $y = -\frac{1}{4}x + \frac{7}{2}$

Special Lines

• Horizontal Line:

  • Slope: $0$

  • Equation: $y = k$

  • Example: $y = 6$

• Vertical Line:

  • Slope: Undefined

  • Equation: $x = k$

  • Example: $x = 4$

Equations of Circles

• Standard Form:

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

  • Where:

    • $(h,k)$ = coordinates of the center point.

    • $r$ = radius of the circle.

• Generating a Circle Equation from Center and Radius:

  • Example: Center = $(3,-2)$, Radius = $4$.

  • Substitute: $(x - 3)^2 + (y - (-2))^2 = 4^2$

  • Resulting Equation: $(x - 3)^2 + (y + 2)^2 = 16$

• Identifying Center and Radius from an Equation:

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

  • Center: $(4,-1)$

  • Radius: $\sqrt{49} = 7$

• Generating a Circle Equation from center and a Point:

  • Example: Center = $(1,2)$, Point on circle = $(4,6)$.

  • Step 1: Calculate radius using the distance formula.

    • $r = \sqrt{(4 - 1)^2 + (6 - 2)^2}$

    • $r = \sqrt{9 + 16}$

    • $r = \sqrt{25} = 5$

  • Step 2: Write Equation.

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

• Circle Equation from Diameter Endpoints:

  • Example: Endpoints $A(0,0)$ and $B(6,8)$.

  • Step 1: Find the Center (Midpoint).

    • $Center = (\frac{0 + 6}{2} , \frac{0 + 8}{2})$

    • $Center = (3,4)$

  • Step 2: Find the Diameter and Radius.

    • $d = \sqrt{(6 - 0)^2 + (8 - 0)^2}$

    • $d = \sqrt{36 + 64} = \sqrt{100} = 10$

    • $Radius = \frac{d}{2} = \frac{10}{2} = 5$

  • Step 3: Write Equation.

    • Answer: $(x - 3)^2 + (y - 4)^2 = 25$

• Converting General Form to Standard Form:

  • Example: Given $x^2 + y^2 - 2x + 6y + 1 = 0$

  • Step 1: Group terms and move constant.

    • $(x^2 - 2x) + (y^2 + 6y) = -1$

  • Step 2: Complete the square.

    • For $x$: $x^2 - 2x + 1 = (x - 1)^2$

    • For $y$: $y^2 + 6y + 9 = (y + 3)^2$

  • Step 3: Balance the equation and finalize.

    • Add both added values ($1$ and $9$) to the right side: $(x - 1)^2 + (y + 3)^2 = -1 + 1 + 9$

    • Standard Form: $(x - 1)^2 + (y + 3)^2 = 9$

    • Center: $(1,-3)$

    • Radius: $3$

Real-Life Applications

• Cell Tower Coverage:

  • Scenario: A tower is located at $(3,4)$ with a coverage radius of $10\text{ km}$.

  • Equation: $(x - 3)^2 + (y - 4)^2 = 100$

• Weather Radar:

  • Scenario: A radar station sits at $(3,-2)$ with a detection range of $40\text{ miles}$.

  • Equation: $(x - 3)^2 + (y + 2)^2 = 1600$

• Cell Phone Plan Cost Modeling:

  • Parameters: Monthly fee = $\$20$, Cost per minute = $\$0.05$.

  • Equation: $C = 0.05m + 20$

Relationships Between Lines and Circles

There are three primary geometric relationships based on the distance ($d$) from the circle's center to the line versus the radius ($r$) of the circle:

1. Secant: Distance < Radius. The line intersects the circle at two points.

2. Tangent: Distance = Radius. The line intersects the circle at exactly one point.

3. Outside: Distance > Radius. The line does not intersect the circle.

• Distance Formula from a Point $(x_0, y_0)$ to a Line $Ax + By + C = 0$:

  • Formula: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$

• Relationship Example:

  • Circle: $x^2 + y^2 = 25$ (Center = $(0,0)$, Radius = $5$)

  • Line: $3x + 4y - 25 = 0$

  • Calculation: $d = \frac{|3(0) + 4(0) - 25|}{\sqrt{3^2 + 4^2}}$

  • Calculation: $d = \frac{25}{5} = 5$

  • Conclusion: Since $d = r$, the line is TANGENT to the circle.

Graphing and Solving Systems

Graphing Systems Checklist

1. Graph the line.

2. Graph the circle.

3. Locate intersection points.

4. Shade required regions if inequalities are involved.

Solving Systems of Equations

• A. Two Lines:

  • Lines: $y = 2x + 1$ and $y = -x + 7$

  • Set equations equal: $2x + 1 = -x + 7$

  • Solve for $x$: $3x = 6 \rightarrow x = 2$

  • Solve for $y$: $y = 2(2) + 1 = 5$

  • Solution: $(2,5)$

• B. Line and Circle:

  • System: $x^2 + y^2 = 25$ and $y = 4$

  • Substitute $y$: $x^2 + 16 = 25$

  • Solve for $x$: $x^2 = 9 \rightarrow x = \pm 3$

  • Solutions: $(3,4)$ and $(-3,4)$

• C. Two Circles:

  • System: $x^2 + y^2 = 25$ and $(x - 4)^2 + y^2 = 25$

  • Expand the second equation: $x^2 - 8x + 16 + y^2 = 25$

  • Subtract the first equation from the expanded second: $-8x + 16 = 0$

  • Solve for $x$: $x = 2$

  • Substitute $x$ into the first circle: $2^2 + y^2 = 25 \rightarrow 4 + y^2 = 25 \rightarrow y^2 = 21$

  • Solve for $y$: $y = \pm \sqrt{21}$

  • Solutions: $(2, \sqrt{21})$ and $(2, -\sqrt{21})$

Formula Sheet

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-Slope: $y - y_1 = m(x - x_1)$

• Slope-Intercept: $y = mx + b$

• Midpoint: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$

• Distance (between points): $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Circle (Standard Form): $(x - h)^2 + (y - k)^2 = r^2$

• Parallel Condition: $m_1 = m_2$

• Perpendicular Condition: $m_1 \times m_2 = -1$

• Distance (Point to Line): $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$

Most Common Exam Questions

Expect questions that require you to:

1. Find the slope given two points.

2. Find the equation of a line passing through two specific points.

3. Find the equation of a line from a given point and slope.

4. Determine a line parallel to a given line.

5. Determine a line perpendicular to a given line.

6. Identify the center and radius of a circle from its equation.

7. Write the equation of a circle given geometric parameters.

8. Find the equation of a circle using diameter endpoints.

9. Convert a circle equation from general form to standard form.

10. Solve systems consisting of two lines.

11. Solve systems consisting of a line and a circle.

12. Solve systems consisting of two circles.

13. Determine if a line is a tangent, secant, or outside of a circle.

14. Model real-life situations using linear or circular equations.

Practice Problems

1. Find the equation of the line through $(-1,4)$ and $(3,-2)$.

2. Find the equation of a circle with center $(-3,1)$ and radius $6$.

3. Find the equation of a circle with diameter endpoints $(2,-1)$ and $(8,5)$.

4. Is $y = 3x + 1$ parallel to $y = 3x - 4$?

5. Find the equation of the line perpendicular to $y = -\frac{2}{3}x + 5$ that passes through $(6,-1)$.

6. Convert $x^2 + y^2 + 4x - 6y - 3 = 0$ to standard form.

7. Find the intersection of $y = 2x$ and $(x - 1)^2 + (y + 2)^2 = 10$.

8. Determine whether $x^2 + y^2 = 16$ and $y = 5$ have zero, one, or two intersections.

9. Find the tangent line to $(x - 1)^2 + (y + 2)^2 = 9$ at point $(4,-2)$.

10. Analyze the relationship between $x^2 + y^2 = 36$ and $6x + 8y - 60 = 0$.