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CIRCLES
- A set of points in the xy-plane equidistant form a fixed point. We called the fixed point as the center and the fixed distance as the radius. Exercise 1. A. Identify the radius and center of the equation. If otherwise, specify if the graph is a single point ( and what this point is), or if it has no point at all.
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Sol: *Group the terms with the same variables together. Transpose the constant. Perform PST on the grouped variables and perform APE (Addition property of equality).
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Sol: To find x-intercept/s, set y to
To find y-intercept/s, set x to
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Sol: *Divide the coefficient of the variables to the terms of the equation to cancel it out.
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Sol: B. Find the equation in standard form of the circle being described in each item. 15.Center at , radius 11 Sol: 16.Center at (6, -5), radius 4 Sol: 17.Diameter with endpoints (-1, 3) and (9, 1) Sol: *Find the midpoint of the diameter by taking the sum of the x and y components of the two points and dividing it by 2. 18.Diameter with endpoints (-2, -1) and (6, 3) Sol: 19.Diameter with endpoints (3, -5) and (6, 4)
42.center at (23, -1), externally tangent to 43.center at (23, -1), internally tangent to 44.tangent externally to both and 45.tangent internally to both and C. Find the equation of the lines described. Give your answers in the form. 46.Tangent line to at points where x = 4 47.Tangent line to at points where x = 15 48.Tangent line to at points where y = 5 49.Tangent line to at points where y = 4 D. For any three noncollinear points, there is exactly one circle that passes through all them. Thus, if we are given three such points, it should be possible to find an equation for the circle. 50.A circle passes through P(5, 9), Q(11, - 3), and R(13, 3). Find its standard form by doing the following steps. a. Determine an equation of the perpendicular bisector of the segment PR. b. Determine an equation of the perpendicular bisector of the segment QR. c. The perpendicular bisector of any chord of a circle will pass through the center. Thus, find the intersection of the two perpendicular bisectors above. This will be the center C of the circle. d. To find r, find the distance of C from any of the three points. e. What is the std. equation of the circle? 51.When expanded, the standard form of the equation of a circle can be written
in , which is the general form of its equation. a. Substitute the coordinates of P(5, 9) into this general form, yielding an equation in A, B, and C. Next, substitute the coordinates Q(11, -3) and R(13, 3), yielding to more equations. Solve these three equations for A, B, and C. b. Using these values of A, B, and C, convert the equation to standard form. Is your answer the same as the one in the preceding item? E. Find the equation of the circle in standard form passing through the three given points. 52.P(1, 1), Q (-3, 9), R(-5, 3) 53.A(3, 6), B(-5, 10), C(-2, 1) 54.M(11, 5), N(6, 10), P(3, 1) 55.Find the values of the constant k so that the graph of is a a) circle, b) a single point, (c) the empty set. 56.Find the values of the constant k so that the graph of is a a) circle, b) a single point, (c) the empty set.
