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Concept explainers
54. The tangent line to a circle may be defined as the line that intersects the circle in a single point, called the point of tangency. See the figure.
If the equation of the circle is and the equation of the tangent line is , show that:
a.
[Hint: The
b. The point of tangency is .
c. The tangent line is perpendicular to the line containing the center of the circle and the point of tangency
![Check Mark](/static/check-mark.png)
To find:
a.
b. The point of tangency is
c. The tangent line is perpendicular to the line containing the center of the circle and the point of tangency.
Answer to Problem 54AYU
Solution:
See explanation.
Explanation of Solution
Given:
The equation of the circle is and the equation of the tangent line .
By definition, the tangent line meets the circle at only one point.
This means that the following system of equations will have only one solution.
-----(1) and -----(2)
Using equation (2), we can replace with in the equation (1)
Therefore
Rewrite this as a standard quadratic equation
-----(3)
We know that, a quadratic equation of the form of will have one solution if and only if its discriminant .
Therefore we can write
Therefore
That is
Divide on 4 on bothsides we get
-----(4)
(b)from equation (3)
We know that the solution of the quadratic equation is
Therefore the coordinate of the point of tangency is
Using equation (4)
Therefore
Substitute this in
That is
Using equation (4),
Hence, prove that the tangency of the point
(c) We know that the slope of the tangent is .
Slope of the radius is slope of the line joining the origin and the point of tangency.
From (b),we know that the tangency is
Therefore
Product between slopes of the radius and the tangent line is .
Hence proved that the two lines are perpendicular.
Chapter 1 Solutions
Precalculus Enhanced with Graphing Utilities
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