Circle Equations and Sketching Practice
Practice drawing circles on a coordinate plane and creating equations for circles with various radii and centers. Understand the relationship between the general equation of a circle and its graph. Explore how different center points and radii affect the appearance of circles. Additionally, learn about the characteristics of circles, such as radius and center point.
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Presentation Transcript
Starter Draw axes from -10 to 10, then copy and complete the table below, to sketch a graph for x + y = 25. x -5 -4 -3 0 3 4 4 5 ?1 ?2 - 4 3 + y = 25 9 + y = 25 y = 25 y = 16 y = 4 or -4
Starter x -5 0 -4 3 -3 -3 4 -4 0 5 3 4 4 3 5 0 ?1 ?2 -5 - 4 -3 x + y = 25 What is the radius of the circle? 5 units 25 = 5 x x x x x x x xx x x x
General Equation of a Circle x + y = r x Radius r, centre (0, 0) x x x x x x xx x x (x a) + (y b) = r x Radius r, centre (a, b) x + y = 25
Can you write an equation that will produce a circle radius 6? Can you write an equation that will produce a circle radius 2? Can you write another equation that will produce a circle radius 2? Write an equation that gives a circle with centre (3, 9)? Write an equation that gives a circle with centre (3,9) AND has a radius of 9?
Worksheet Centre (0, 0) Centre (3, 4) Centre (-1, 5) Centre (___, ___) Radius 4 Radius 6 Radius ___ (x + 1) + (y 5) = 49 Radius 5 (x + 2) + (y + 1) = 25 Radius ___ x + y = 1 Extension: Can you sketch these circles? Make sure your axes are an appropriate size
Answers Centre (0, 0) Centre (3, 4) Centre (-1, 5) Centre (___, ___) Radius 4 x + y = 16 (x 3) + (y 4) = 16 (x + 1) + (y 5) = 16 (x + 2) + (y + 1) = 16 Radius 6 x + y = 36 (x 3) + (y 4) = 36 (x + 1) + (y 5) = 36 (x + 2) + (y + 1) = 36 Radius 7 (x + 1) + (y 5) = 49 x + y = 49 (x 3) + (y 4) = 49 (x + 2) + (y + 1) = 49 Radius 5 (x + 2) + (y + 1) = 25 x + y = 25 (x 3) + (y 4) = 25 (x + 1) + (y 5) = 25 Radius 1 x + y = 1 (x 3) + (y 4) = 1 (x + 1) + (y 5) = 1 (x + 2) + (y + 1) = 1
Recap y = mx + c c is the y- intercept, or where the graph cuts the y-axis m is the gradient, or the slope of the graph Gradients of perpendicular graphs sum to -1. (negative reciprocals)
Gradient of line 1 Gradient of its perpendicular line 1 3 5 6 7 - 3 9 1 - 8 6 1 9 1 8 1 1 5 7
Step-by-step guide to calculate the equation of a tangent of a circle at a given point: 1. Calculate the gradient of the radius of the circle. Gradient of the radius the tangent x -1 = Gradient of 2. Calculate the gradient of the tangent of the circle. 3. Substitute the given coordinate and the gradient of the tangent into y = mx + c to calculate the y- intercept.
Example Find the equation of the tangent to the circle at the point (3, 4). = -3 4 Gradient of radius = rise = 4 3 Gradient of tangent = -1 4 = -1 x 3 4 run 3 y = mx + c 4 = -3 x 3 + c 4 4 = -9 + c 4 25 = c 4 y = -3x + 25 4 4 x
Answers y = -3x + 3 4 y = -4x + 10 3 3 y = -4x + 9 3 y = x + 10 y = 4x 40 3 3 y = -9x 145 8 8 y = 12x + 12 5 x = 3
