Chapter 10.4, Problem 24PSC

To determine

To Prove : Center of the circle.

Answer to Problem 24PSC

  $\text{Center : (13, 12) or (13, }-12\text{)}$

Explanation of Solution

Given information :

  $\text{Center of a circle is tangent to the y-axis and intersects the x-axis at (8, 0) and (18, 0)}\text{.}$

  

Since the circle is touching the y-axis the center must be midway between the points where it cross the x-axis.

  $\begin{array}{l}\text{So, x-coordinate of center = 13}\\ \text{Because the circle is tangent to the y-axis, so radius is 13}\text{.}\\ \text{Length of the chord = 10}\\ \text{Half of the length of chord = 5}\\ \text{So, Radius = 13, half of chord = S}\\ \text{Using Pythagoras, distance from the center to the chord is 12}\text{.}\\ \text{[}\because {\text{13}}^{\text{2}}-5^2{\text{=12}}^{\text{2}}\text{]}\\ \text{So, center of the circle is 12 units from the x-axis}\text{.}\\ \text{Since it could be above or below}\text{.}\\ \therefore \text{Center is either (13, 12) or (13, }-12\text{)}\text{.}\end{array}$