Equations for Tangent Lines to Circles

New York State Common Core Math Geometry, Module 5, Lesson 19

Worksheets for Geometry

Student Outcomes

• Given a circle, students find the equations of two lines tangent to the circle with specified slopes.
• Given a circle and a point outside the circle, students find the equation of the line tangent to the circle from that point.

Equations for Tangent Lines to Circles

Classwork

Opening Exercise

A circle of radius 5 passes through points 𝐴(−3, 3) and 𝐵(3, 1).
a. What is the special name for segment 𝐴𝐵?
b. How many circles can be drawn that meet the given criteria? Explain how you know.
c. What is the slope of 𝐴𝐵?
d. Find the midpoint of 𝐴𝐵.
e. Find the equation of the line containing a diameter of the given circle perpendicular to 𝐴𝐵.
f. Is there more than one answer possible for part (e)?

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Example 1

Consider the circle with equation (𝑥 − 3)2 + (𝑦 − 5)2 = 20. Find the equations of two tangent lines to the circle that each has slope 1/2.

Example 2

Refer to the diagram below.
Let 𝑝 > 1. What is the equation of the tangent line to the circle 𝑥2 + 𝑦2 = 1 through the point (𝑝, 0) on the 𝑥-axis with a point of tangency in the upper half-plane?

Exercises 2–4

2. Use the same diagram from Example 2 above, but label the point of tangency in the lower half-plane as 𝑄′.
   a. What are the coordinates of 𝑄′?
   b. What is the slope of 𝑂𝑄′?
   c. What is the slope of 𝑄′𝑃?
   d. Find the equation of the second tangent line to the circle through (𝑝, 0).
3. Show that a circle with equation (𝑥 − 2)2 +(𝑦 + 3)2 = 160 has two tangent lines with equations 𝑦 + 15 = 1/3 (𝑥 −6) and 𝑦 − 9 = 1/3(𝑥 + 2).
   
4. Could a circle given by the equation (𝑥 − 5)2 + (𝑦 − 1)2 = 25 have tangent lines given by the equations 𝑦 − 4 = 4/3 (𝑥 − 1) and 𝑦 − 5 = 3/4 (𝑥 −8)? Explain how you know.

Lesson Summary

THEOREM:

A tangent line to a circle is perpendicular to the radius of the circle drawn to the point of tangency.

Relevant Vocabulary

TANGENT TO A CIRCLE: A tangent line to a circle is a line in the same plane that intersects the circle in one and only one point. This point is called the point of tangency.

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