Problem 40 - Algebra 2

Question

Write the equation of the circle that passes through the given point and has a center at the origin. (Hint: You can use the distance formula to find the radius.) $$ (-5,-12) $$

Step-by-Step Solution

Answer

The equation of the circle that passes through the point (-5, -12) and has a center at the origin is \( x^2 + y^2 = 169 \).

Step 1: Use the Distance Formula to Determine the Radius

The distance formula is derived from Pythagoras’ theorem and is used to calculate the distance between two points in a coordinate system. The formula is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Given that our two points are the origin (0,0) and (-5, -12), we can plug in these values to calculate the radius of the circle, \( r \). This gives us \( r = \sqrt{(-5 - 0)^2 + (-12 - 0)^2} \). Calculating this, we find that \( r = 13 \).

Step 2: Formulating the Equation of the Circle

The general equation of a circle with its center at the origin is \( x^2 + y^2 = r^2 \). We have found that r is equal to 13 for our circle. Substituting this radius into the equation, we get the equation of the circle as \( x^2 + y^2 = 13^2 \).

Step 3: Simplifying the Equation

Simplifying \( 13^2 \), we can finalize our equation as \( x^2 + y^2 = 169 \).

Key Concepts

Understanding the Distance Formula

The distance formula is an essential tool in coordinate geometry. It helps to determine the distance between two points on the coordinate plane. Think of it as a way to measure the "straight-line" distance, the same as if you would use a ruler in the physical world. The formula is:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

This may look a bit complex, but it's based on the Pythagorean theorem. If you imagine a right triangle where the two points form the ends of the hypotenuse, the formula calculates the length of that hypotenuse. In using this formula, just plug in the x and y coordinates of the two points and solve.
In our problem, we use the origin point (0,0) and the given point (-5,-12). Substituting these into the formula gives:

• \( r = \sqrt{(-5 - 0)^2 + (-12 - 0)^2} \)
• \( r = \sqrt{25 + 144} \)
• \( r = \sqrt{169} \)
• \( r = 13 \)

Here, 13 is the radius of our circle.

Basics of Coordinate Geometry

Coordinate geometry, also known as analytic geometry, marries algebra and geometry to describe geometric figures using equations. In this plane, any point is defined by an ordered pair of numbers, \((x, y)\). This system helps us visualize algebraic equations and simplify geometric problems through algebra.
In our exercise, we need to visualize a circle which has its center at the origin, (0,0). Circles in coordinate geometry are unique because they benefit from a simple equation when centered at the origin. It is given by:

• \( x^2 + y^2 = r^2 \)

where \( r \) is the radius of the circle. If the center were not at the origin, the equation would be slightly more complex.
Using coordinate geometry simplifies our task of finding the circle's equation and allows us to connect algebra with geometry seamlessly.

Steps for Radius Calculation

Calculating the radius is often the first step in forming the circle's equation. The radius of a circle is the distance from the center of the circle to any point on its edge. By using the distance formula with one endpoint at the circle's center and the other at any point on the circle's circumference, we obtain the radius.
In this exercise, since the circle is centered at the origin and passes through the point (-5, -12), we use the coordinates of these points in the distance formula:

• \( r = \sqrt{(-5 - 0)^2 + (-12 - 0)^2} \)
• \( r = \sqrt{25 + 144} \)
• \( r = \sqrt{169} \)
• \( r = 13 \)

This tells us that the radius of the circle is 13. With this radius, we can directly form the equation of the circle as \( x^2 + y^2 = 169 \), thanks to the elegance of the circle centered at the origin on the coordinate plane.