Problem 6 - The Calculus with Analytic Geometry

Question

In Exercises 5 through 10, find an equation of the circle satisfying the given conditions. Center is at \((-2,5)\) and tangent to the line \(x=7\).

Step-by-Step Solution

Answer

(x+2)^2 + (y-5)^2 = 81

Step 1: Determine the radius

The radius of the circle is the distance from the center of the circle (-2,5) to the tangent line x=7. This distance is simply the horizontal distance between x-coordinates of the center and the line. So, the radius = |-2 - 7| = 9.

Step 2: Formulate the equation of the circle

The standard equation for a circle with center (a,b) and radius (r) is (x-a)^2 + (y-b)^2 = r^2. Substituting a=-2, b=5 and r=9, we get:(x+2)^2 + (y-5)^2 = 81

Key Concepts

tangent line

A tangent line is a straight line that touches a circle at exactly one point. This point is called the point of tangency. The tangent line is perpendicular to the radius of the circle at this point.

To determine the radius using a tangent line, you need to calculate the perpendicular distance from the circle's center to the tangent line. In our example, the circle is tangent to the line x = 7.
Hence, the center of the circle is at (-2, 5), and the line touches the circle exactly once at x = 7. By measuring the distance along the x-axis, we find the radius to be | -2 - 7 | = 9.

radius calculation

The radius of a circle is the distance from its center to any point on its circumference. This can be computed using different geometric principles.

For this problem, the given center is at (-2, 5), and the circle is tangent to the line x = 7. To find the radius, we look for the horizontal distance from the center to the tangent line.

• Calculate the absolute value of the difference between the x-coordinates of the center (-2) and the tangent line (7).
• So, radius = | -2 - 7 | = 9.

With the radius known, we can proceed to formulate the circle's equation.

standard equation of a circle

The standard equation of a circle with a center at (a, b) and radius r is given by: \[ (x - a)^2 + (y - b)^2 = r^2 \]

To put this into practice, we take the given coordinates of the center (-2, 5) and the radius 9. Substituting these values into the standard form, we get:

• (x + 2)^2 + (y - 5)^2 = 9^2
• (x + 2)^2 + (y - 5)^2 = 81

This equation represents a circle with a center at (-2, 5) and a radius of 9 units, which perfectly satisfies the given conditions.