Understanding Systems of Equations Involving Lines and Circles: Tangents, Secants, and Solutions

Review
Recall the following terms.
• circle
• center
• diameter
• radius
• circumference
• chord
• quadratic equation
Racel N. Pilloses, MSc
Notre Dame of Marbel University – IBED SHS
System of Equations involving Lines and Circles

Review
Equation of a Line:
Standard Equation of a Circle:

System of Equations involving
Lines and Circles
RACEL N. PILLOSES, LPT, MSc
Pre-Calculus Teacher

Learning Objectives
At the end of the lesson, the learners should be able to:
• determine the geometric relationships between a line and a circle
by analyzing their equations;
• solve systems of equations involving lines and circles using the
substitution method;
• graph the solution set of systems of equations involving lines and
circles on the coordinate plane using GeoGebra;

Learning Objectives
At the end of the lesson, the learners should be able to:
• solve real-life problems involving the relationships between lines
and circles using appropriate equations and graphical
representations; and
• appreciate the lesson through citing real-life applications of the
relationships of lines and circles.

Question
What do you call a line that intersects a circle at
exactly one point?
What do you call a line that intersects a circle at
exactly two points?

Tangent of a Circle
 A tangent of a circle is a
straight line that touches the
circumference of the circle at
only one point.
 The angle between a tangent
and radius is 90 degrees.
 The tangent line is
perpendicular to the circle's
radius at the point of tangency.
Illustration:

Tangent Theorems
1. Radius-Tangent Theorem
The angle between a tangent
and radius is 90 degrees.
2. Two-Tangent Theorem
Tangents that meet at the same
point are equal in length.

Secant of a Circle
 A secant of a circle is a line
that intersects a circle at
exactly two points.
 Secant is derived from the
Latin word secare, which
means to cut.
 It can also be understood as
the extension of the chord of
a circle that goes outside the
circle.
Illustration:

Question
How to determine if a line is tangent or secant
to a circle using its equations?

Systems of Equations
The most common method for finding the intersection (solution) is the
substitution method.
1. Start with the equations: You'll have the equation of the line and the
equation of the circle ().
2. Substitute: Substitute the expression for from the line equation into
the circle equation. This will give you a quadratic equation in terms
of only.
3. Solve the quadratic equation: You can use the quadratic formula,
factoring, or completing the square to solve for .
4. Find the y-coordinates: Once you have the values, substitute them
back into the equation of the line to find the corresponding values.

The illustrations show the possible solution sets for a system of equations involving
a circle and a line.
• No solution. The line does not intersect the circle.
• One solution. The line is tangent to the circle and intersects the circle at exactly one point.
• Two solutions. The line crosses the circle and intersects it at two points.

Instruction: Solve and determine the solution set of the system of
equations involving a line and a circle.
Example 1:
and

Example 1:
Solution:
|
|
Solution: (continuation…)
@
@

Example 1:
and
Conclusion:
• The two solutions or points of intersection are and .
• Based on the result, we can say that the line is a secant line of the
circle since we have two points of intersection.
• We can verify our answer using the graphs in GeoGebra.

Graph in GeoGebra
(𝟑,𝟒)
(−𝟒,−𝟑)

Example 2:
and

Example 2:
Solution:
@

Example 2:
and
Conclusion:
• The solution or point of intersection is .
• Based on the result, we can say that the line is a tangent line of the
circle since we have one point of intersection.

Graph in GeoGebra
(𝟑,𝟏)

Example 3:
and

Example 3:
Solution:
Complete the squares:
is an imaginary number.

Example 3:
and
Conclusion:
• There is NO solution.
• Based on the result, we can say that there is no point of intersection.
Hence, there is no solution.

Graph in GeoGebra

Using Discriminants
You can identify the number of solution/s using its discriminants.
Solve for discriminant:
Interpretation:

Using Discriminants
Let’s use the equations from the previous examples.
1.
Therefore, there are two solutions.
2.
Therefore, there is one solution.

Board work Activity
Instruction: Determine the solution set using the substitution method.

Board work Activity
Instruction: Determine the solution set using the substitution method.
Hint: Use the quadratic equation.

Board work Activity
Group 1
Group 2

 A tangent of a circle is a straight line that touches the circumference
of the circle at only one point.
 The angle between a tangent and radius is 90 degrees.
 The tangent line is perpendicular to the circle's radius at the point of
tangency.
 A secant of a circle is a line that intersects a circle at exactly two
points.
 Secant is derived from the Latin word secare, which means to cut.
 It can also be understood as the extension of the chord of a circle
that goes outside the circle.
Summary

Instruction: Find the solution set of the given set of equations. Write
your answers on a 1 whole sheet of paper. (20 points)
1. 3.
2. 4.
Seatwork

A. Multiple Choice. Read the statements/questions carefully and
choose the letter of the correct answer. Write the CAPITAL LETTERS.
(10 points)
1. A line that intersects a circle at exactly one point is called a
__________.
A. chord
B. secant
C. tangent
D. diameter
Quiz 3

2. A line that intersects a circle at exactly two points is called
a ___________.
A. tangent
B. secant
C. radius
D. diameter
Quiz 3

3. The radius drawn to the point of tangency is always ________ to
the tangent line.
A. parallel
B. perpendicular
C. equal
D. opposite
Quiz 3

4. A system consisting of a secant line and a circle has __________.
A. no solution
B. one solution
C. two solutions
D. infinitely many solutions
Quiz 3

5. When solving a system involving a line and a circle using
substitution, the resulting equation is usually a ____________.
A. linear equation
B. exponential equation
C. quadratic equation
D. rational equation
Quiz 3

6. The discriminant of the quadratic equation is positive. This indicates
that the system has ____________.
A. no real solution
B. one real solution
C. two real solutions
D. infinitely many solutions
Quiz 3

7. Determine the number of solutions if the discriminant is 0.
A. No solution
B. One solution
C. Two solutions
D. Four solutions
Quiz 3

8. A line intersects a circle at the points (2,3) and (5,7). The line
is a ____________.
A. tangent
B. secant
C. radius
D. diameter
Quiz 3

9. Determine the number of solutions if the discriminant is −25.
A. No solution
B. One solution
C. Two solutions
D. Three solutions
Quiz 3

10. Which statement correctly describes the solution set of a system
involving a line and a circle?
A. The solution set is always one point.
B. The solution set is always two points.
C. The solution set consists of the center of the circle.
D. The solution set consists of the point(s) where the line and
circle intersect.
Quiz 3

B. Solving. Find the solution set of the given set of equations. Write
your solutions and answers on a 1 whole sheet of paper. (20 points)
1. 3.
2. 4.
Quiz 3

Park Layout Mathematical Modelling
Directions:
1. Create a layout of a city park on a clean sheet of paper or using any
digital drawing application.
2. Your design must include the following:
- At least 4 straight walking paths represented by equations of
lines.
- At least 2 circular fountains or gardens represented by equations
of circles.
- At least one pair of parallel lines.
- At least one pair of perpendicular lines.
Performance Check

Directions:
3. Draw the layout on a coordinate plane and label all important points.
4. Write the equation of every line and circle used in your design.
5. Show the necessary computations used to obtain each equation.
6. Label each feature of the park (e.g., Main Walkway, Bike Lane, Central Fountain,
Garden Circle).
7. Submit the following:
Park layout with labels
Coordinate plane
Equations of all lines and circles
Complete solutions and computations
A brief explanation (100 to 150 words) describing how mathematics was used in
your
design.
Performance Check

Scoring Rubric:
Performance Check
Criteria Excellent (10) Good (8) Satisfactory (6) Needs Improvement (4)
Mathematical Accuracy All equations of lines and
circles are correct.
Computations are complete
and error free.
One minor computational or
equation error.
Two to three errors that do
not greatly affect the design.
Numerous errors or incorrect
equations.
Application of Concepts All required concepts are
correctly applied, including
parallel lines, perpendicular
lines, and circles.
One required concept is
incomplete or slightly
incorrect.
Two required concepts are
incomplete or incorrect.
Several required concepts
are missing.
Layout Design and
Creativity
Layout is realistic, organized,
visually appealing, and
demonstrates originality.
Layout is organized with
good creativity.
Layout is simple but
complete.
Layout is disorganized or
lacks creativity.
Completeness of
Requirements
All required components are
present, labeled, and
explained.
One required component is
missing.
Two required components
are missing.
Several required
components are missing.
Presentation and
Organization
Work is neat, labels are
clear, and solutions are easy
to follow.
Generally neat with minor
issues in organization.
Readable but contains
several organizational
problems.
Difficult to read due to poor
organization or presentation.

Understanding Systems of Equations Involving Lines and Circles: Tangents, Secants, and Solutions