Find the center and the radius of the circle that passes through the points (-2,7),(0,1), and (2

step 1 Hello everyone today we are going to solve problem number 32 here we had to find the equation of

Find the center and the radius of the circle that passes...

Key Concepts

Unique Circle Through Three Non-Collinear Points

A key geometric principle is that exactly one circle can be drawn through any three non-collinear points. This uniqueness provides the basis for solving problems where the goal is to find the center and radius of the circle that intersects all three points, underpinning the method involving perpendicular bisectors or algebraic elimination.

System of Equations

Setting up a system of equations is necessary when plugging the coordinates of points into the circle's equation. Solving this system allows for determining the unknown center coordinates and the radius, showcasing the interplay between algebra and geometry in finding parameters of geometric figures.

Standard Equation of a Circle

This concept involves understanding that the equation of a circle is typically expressed in the form (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is its radius. This form is fundamental for identifying and working with circles in algebra and coordinate geometry.

Distance Formula

The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in the coordinate plane. It is used to compute the radius of a circle by finding the distance between the center and any point on the circle, and it is essential in verifying that all given points lie at an equal distance from the determined center.

Circumcenter of a Triangle

When a circle passes through three non-collinear points, the center of the circle is known as the circumcenter of the triangle formed by those points. This concept involves finding the intersection point of the perpendicular bisectors of the triangle’s sides, a method that is central to solving problems involving circles through given points.

Transcript

Please give Ace some feedback