Problem 41 - 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.) $$ (12,-5) $$

Step-by-Step Solution

Answer

The equation of the circle is \( x^2 + y^2 = 169 \).

Step 1: Find the Radius

Use the distance formula to find the radius, which should be \( r = \sqrt{(12 - 0)^2 + (-5 - 0)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \). The radius r is hence 13 units.

Step 2: Write the Equation of the Circle

Now that the center of the circle and its radius are known, it's possible to write the equation of the circle. In standard form, that's \( x^2 + y^2 = r^2 \). Substituting r = 13 into this equation, the result is \( x^2 + y^2 = 13^2 \), or \( x^2 + y^2 = 169 \).

Key Concepts

Distance Formula

The distance formula is a handy tool used to find the distance between two points on a coordinate plane. If you have two points, \(x_1, y_1\) and \(x_2, y_2\), the formula is given by

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

This formula derives from the Pythagorean theorem, where the distance serves as the hypotenuse of a right-angled triangle formed by the differences in x and y coordinates. When applied in our exercise, the center of the circle is at the origin (0,0), which simplifies the formula for finding the radius of a circle. Simply plug in the coordinates of the given point (12, -5) to get \[ r = \sqrt{(12 - 0)^2 + (-5 - 0)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]. So, the radius is 13 units.

Radius Calculation

Calculating the radius of a circle is straightforward once you know the center and a point on the circle. The radius is simply the distance between these two points. In many problems, especially where the circle is centered at the origin, this becomes incredibly uncomplicated.
For example, with the center at \(0, 0\) and a point on the circle like \(12, -5\), use the distance formula as shown earlier to find

• \( r = \sqrt{(12 - 0)^2 + (-5 - 0)^2} \)
• Calculating this, we find \[ r = \sqrt{169} = 13 \]

Thus, the radius r is 13 units. This step is fundamental in forming the equation of the circle, providing a clear visualization of how large the circle is.

Standard Form of Circle Equation

The standard form of a circle's equation connects its center and radius in a neat algebraic format. Whenever the circle's center is \(h, k\) and the radius is r, this is expressed as

• \((x - h)^2 + (y - k)^2 = r^2\)

However, in our problem, since the center is at the origin (0, 0), the equation becomes more straightforward. It simplifies to \(x^2 + y^2 = r^2\).
With a radius of 13, this equation turns into

• \(x^2 + y^2 = 13^2\)
• Or more simply, \[x^2 + y^2 = 169\]

This form is efficient in expressing all points \(x, y\) that lie on the circle, providing a direct way to understand its geometry on a plane.