Problem 34 - Algebra 2

Question

Write an equation for the circle that satisfies each set of conditions. center at \((-8,-7),\) tangent to \(y\) -axis

Step-by-Step Solution

Answer

The equation of the circle is \((x+8)^2 + (y+7)^2 = 64\).

Step 1: Determine the Radius of the Circle

A circle tangent to the y-axis must have its radius equal to the horizontal distance from its center to the y-axis. Since the center is (-8, -7), the radius of the circle will be 8 units.

Step 2: Write the Equation of the Circle

The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where (h, k) is the center, and r is the radius. For this circle, the center is (-8, -7) and r is 8, so the equation is \((x+8)^2 + (y+7)^2 = 64\).

Key Concepts

Understanding Circle Geometry

Circle geometry is a fascinating part of mathematics that involves understanding various properties of circles. A circle is a set of all points in a plane that are at a fixed distance from a center point. This distance is called the radius.

A few fundamental terms related to circles include:

• Center: The fixed point from which every point on the circle is equidistant.
• Radius: The distance from the center of the circle to any point on its circumference.
• Diameter: The longest distance across the circle, which is twice the length of the radius.
• Circumference: The total distance around the circle’s edge.

In geometry, understanding how a circle interacts with a straight line, such as being tangent to the y-axis, is crucial. A tangent line touches the circle at exactly one point without crossing its boundary.

Radius Calculation for Tangent Circles

Calculating the radius is a key step in understanding the position and size of a circle. When a circle is tangent to an axis, its radius is the distance from its center to the point where it touches the axis.

For a circle tangent to the y-axis, the horizontal distance between the center of the circle and the y-axis gives you the radius. In our exercise, the circle's center is at \((-8, -7)\), and thus, the horizontal distance to the y-axis is \(8\) units. Therefore, the radius \(r\) is \(8\).

Knowing the radius helps us not only to determine the circle's equation but also to visualize the circle's size and position relative to the coordinate plane. It can also give a clue on how close other points or lines are to the circle.

Standard Form Equation of a Circle

The standard form equation of a circle is a powerful tool for describing a circle in the coordinate plane. The general formula is:\[(x-h)^2 + (y-k)^2 = r^2\]Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius. This formula perfectly captures the geometric definition, ensuring that all points \((x, y)\) on the circle are exactly \(r\) units away from the center.

In our example where the center is \((-8, -7)\) and the radius is \(8\), substituting these values into the formula gives:\[(x + 8)^2 + (y + 7)^2 = 64\]This equation can now be used to describe or find other properties of the circle, such as identifying points on its circumference or exploring geometric transformations. Understanding how to form and manipulate this equation is key to mastering circle geometry in algebra.