Find properties of circles

Learning Objectives Identify the center and radius of a circle from its equation in standard form. Convert the equation of a circle from general form to standard form by completing the square. Determine the center and radius of a circle from its equation in general form. Write the equation of a circle given its center and radius, or its center and a point on the circle. Determine if a given point lies inside, on, or outside a circle. Find the equation of a tangent line to a circle at a given point using implicit differentiation. How does your phone's GPS pinpoint your location using signals from satellites? 🛰️ It's all based on the intersection of circles! This tutorial explores the circle, a fundamental conic section. You will learn to analyze and write the equat...

TermDefinitionExample CircleA circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center).All points that are exactly 5 units away from the point (2, 3) form a circle. Center (h, k)The fixed point in the middle of a circle from which all points on the circle are equidistant.In the equation (x - 4)^2 + (y + 1)^2 = 25, the center is (4, -1). Radius (r)The fixed distance from the center of a circle to any point on the circle. It must be a positive value.In the equation (x - 4)^2 + (y + 1)^2 = 25, the radius is \sqrt{25} = 5. Standard Form of a Circle's EquationThe equation of a circle written as (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.(x - 2)^2 + (y - 3)^2 = 16 is the standard form for a circle...

Standard Form Equation of a Circle (x - h)^2 + (y - k)^2 = r^2 Use this form to easily identify the circle's center (h, k) and radius r. This is also the form you aim for when writing the equation of a circle. General Form to Standard Form Conversion x^2 + y^2 + Dx + Ey + F = 0 \rightarrow (x^2 + Dx) + (y^2 + Ey) = -F \rightarrow (x - h)^2 + (y - k)^2 = r^2 To find the center and radius from the general form, group the x-terms and y-terms and use the method of completing the square for each variable. Slope of a Tangent Line (via Implicit Differentiation) \frac{d}{dx}[(x-h)^2 + (y-k)^2] = \frac{d}{dx}[r^2] \rightarrow 2(x-h) + 2(y-k)\frac{dy}{dx} = 0 \rightarrow \frac{dy}{dx} = -\frac{x-h}{y-k} To find the slope of the tangent line at a point (x_1, y_1) on the ci...