Problem 37 - 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.) $$ (-5,0) $$

Step-by-Step Solution

Answer

The equation of the circle that passes through the point (-5,0) and has its center at the origin is \(x^2 + y^2 = 25\).

Step 1: Find the radius

Use the distance formula to calculate the radius of the circle. The distance formula is given by \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\), where \(d\) is the distance, and \((x_1, y_1)\) and \((x_2, y_2)\) are the points. In this case, the origin (0,0) is \((x_1, y_1)\) and (-5,0) is \((x_2, y_2)\). Hence, \(d = \sqrt{(-5-0)^2 + (0-0)^2} = 5\)

Step 2: Write the equation of the circle

The equation of a circle with center (h,k) and radius r is \((x-h)^2 + (y-k)^2 = r^2\). Here, the center is at the origin, hence h=0, k=0 , and r=5. So substituting these values in the equation, we get the equation of the circle as \(x^2 + y^2 = 25\)

Key Concepts

Distance Formula

To fully understand how to find the radius of a circle that passes through a certain point with a known center, you first need to grasp the concept of the distance formula. The distance formula is a powerful mathematical tool used to calculate the length of the line segment between two points in a coordinate plane. The formula is expressed as: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]In this formula, - \(d\) is the distance between the two points,- \(x_1, y_1\) are the coordinates of the first point, and- \(x_2, y_2\) are the coordinates of the second point. In practice, this means you subtract the x-coordinates and y-coordinates of the two points respectively, square these differences, and then take the square root of the sum of these squares. This computation gives you the straight-line distance between the two points. It’s a direct application of the Pythagorean theorem in a coordinate plane.

Radius of a Circle

The radius of a circle is a crucial concept in geometry. It is defined as the distance from the center of the circle to any point on its circumference. Understanding how to calculate the radius is key to solving problems involving circles. In our example, once we identify a specific point on the circle and know the circle's center, we can use the distance formula to get the radius. In this problem, we calculate the radius with the point \((-5, 0)\) and center \(0, 0\). Using the distance formula gives us:\[d = \sqrt{(-5-0)^2 + (0-0)^2} = \sqrt{25} = 5\]Thus, the radius is 5. Remember:- The radius is always a positive number.- Every point on the circle is equidistant from the center, meaning that if you select another point on the circle, the calculated radius will remain the same.

Center of a Circle

The center of a circle is the fixed point equidistant from all points on the circle's circumference. Understanding its role is essential when working with the equation of a circle. In this instance, the center is provided as the origin \(0, 0\). The equation of a circle with a given center \((h, k)\) and radius \(r\) is:\[(x-h)^2 + (y-k)^2 = r^2\]For our circle centered at \(0, 0\), the equation simplifies. We substitute \(h = 0\) and \(k = 0\) into the equation, resulting in:\[x^2 + y^2 = r^2\]With the radius previously calculated as 5, the equation becomes:\[x^2 + y^2 = 25\]This form is particularly simple and shows that all points \((x, y)\) on the circle will satisfy this equation.