Problem 43 - 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.) $$ (1,-5) $$

Step-by-Step Solution

Answer

The equation of the circle that passes through the point (1,-5) and has a center at the origin is \(x^{2} + y^{2} = 26\).

Step 1: Find the radius

Use the distance formula to find the radius of the circle. The radius is the distance between the center of the circle (0,0) and the point (1,-5). The distance formula \(\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\) can be simplified to \(\sqrt{(1-0)^{2} + ((-5) - 0)^{2}}\) which equals \(\sqrt{1^2 + (-5)^2}\) or \(\sqrt{26}\)

Step 2: Write down the equation of the circle

Now that we have the radius, we can use the formula for the equation of a circle to form the equation with \(r^2 = 26\): \(x^{2} + y^{2} = 26\).

Key Concepts

Distance Formula

The distance formula is a pivotal concept in coordinate geometry to calculate the distance between two points in a plane. It is derived from the Pythagorean theorem. To find the distance between points \((x_1, y_1)\) and \((x_2, y_2)\), use this formula: \[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This calculates the straight-line distance between the two points, often referred to as the Euclidean distance. In this exercise,

• Point 1 is the center of the circle: \((0,0)\)
• Point 2 is the given point on the circle: \((1,-5)\)

Applying these coordinates into the formula, we initially simplify to find the radius, the length of which is critical for defining the circle's equation.

Radius

The radius of a circle is the distance from its center to any point on its circumference. In our case, since one point on the circle and the center at the origin are known, the distance formula comes into play. By plugging the coordinates of the center \((0,0)\) and the point \((1,-5)\) into the distance formula:\[\text{Radius} = \sqrt{(1 - 0)^2 + (-5 - 0)^2} = \sqrt{1^2 + (-5)^2} = \sqrt{26}\]The radius here is not a whole number, but rather a square root measurement, \(\sqrt{26}\). This value is crucial as it reflects the constant distance from the center to the point on the circle.

Center of a Circle

The center of a circle is a fixed point equidistant from all points on the circle's edge. In this exercise, the center is positioned at the origin, \((0,0)\). This simplifies our calculations significantly, as most components of the distance formula simplify to simple arithmetic. When the center is at the origin, the generic equation of the circle follows the pattern:\[x^2 + y^2 = r^2\]With the origin as the center, establishing the circle's geometry becomes straightforward, facilitating easy determination of its equation once the radius is known.

Coordinate Geometry

Coordinate geometry, often termed analytic geometry, merges algebra and geometry using a coordinate system. It enables the representation of geometric shapes in a numerical format. For circles, the fundamental equation is:\[(x - h)^2 + (y - k)^2 = r^2\]where \((h, k)\) represents the circle's center, and \(r\) is its radius. This equation beautifully encapsulates the continuous nature and symmetry of a circle. In this particular problem, the circle's center is at \((0,0)\), thus simplifying the equation to:\[x^2 + y^2 = r^2\]Given that we've calculated \(r^2 = 26\), the equation of the circle simplifies even further to:\[x^2 + y^2 = 26\]This formulation provides a complete description of the circle in the coordinate plane, highlighting the elegance of coordinate geometry in defining shapes mathematically.