Understanding Circle Equations: Centre, Radius, and Equations of Circles
Learn how to find the centre and radius of a circle, as well as equations related to circles. Understand concepts such as diameter, coordinates, and equations through examples and explanations.
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Circle Equations
Circle Equations BAT Find and use equations of circles BAT find the centre and radius of a circle from its equation KUS objectives Starter: AB is the diameter of a circle. A is (2, -5) B is (-3, 2) Find: 1 2, 3 The centre of the circle 2 1 2 52+ 72=1 The radius of the circle 74 2 The equation of line AB Gradient 7 ? + 5 = 7 7? + 5? + 11 = 0 5 5(? 2) or
Notes 1 The equation of the circle with centre (5,7) and radius 4 This is a translation of ?2+ ?2= 42 (5,7) This is a translation 5 units to the right This is a translation 7 units up
? ??+ ? ??= ?? ?????? = (?,?) ?????? = ? Notes 2 Find the coordinates of the centre, and the radius of, the circle with the following equation ? + 32+ ? 12= 42 Remember the coordinates have the opposite sign to what is in the brackets! ?????? = ( 3,1) ?????? = 4 If the equation is written as above, the radius is obvious!
WB35Show that the circle: ? 32+ ? + 42= 20 Passes through (5, -8) ? 32+ ? + 42= 20 (5) 32+ ( 8) + 42= 20 22+ 42= 20 4 + 16 = 20 As the statement is true, the circle curve will pass through (5,-8)
Find the coordinates of the centre, and the radius of, the circle with the following equation: WB36 2 ? 5 + ? + 42= 32 ?) 2 Remember the coordinates have the opposite sign to what is in the brackets! 5 2, 4 ?????? = ?????? = 32 = 4 2 If the equation is written as above, square root the number ? + 32+ ? 62 49 = 0 ?) ?????? = 3,6 ?????? = 49 = 7
The line AB is the diameter of a circle, where A and B are (4,7) and (-8,3) respectively. Find the equation of the circle. To do this we need to find the centre of the circle, and its radius WB37 Finding the midpoint (4,7) (-2,5) 8 + 4 2 ,3 + 7 2 = 2,5 (-8,3) Finding the radius Using (-2,5) and (4,7) 4 ( 2)2+ 7 52 = 40 = 2 10 Equation of the circle ? + 22+ ? 52= (2 10)2 ? + 22+ ? 52= 40
WB38 The line 4x 3y 40 = 0 touches the circle (x 2)2 + (y 6)2 = 100 at P = (10, 0) Show that the radius at P is perpendicular to this line. (x 2)2 + (y 6)2 = 100 4x 3y 40 = 0 To solve this problem, you need to find the gradient of the straight line and compare it to the gradient of the radius at (10,0) (2,6) 4? 3? 40 = 0 4 3? 40 (10,0) 3= ? So the gradient of the straight line is 4 3 Now find the gradient of the radius at (10,0) Multiply the gradients together 4 3 3 = 12 0 (6) (10) (2)= 3 ? = 4 12 4 = 1 So this is the gradient of the radius at (10,0) If two gradients multiply to make -1, they are perpendicular
WB 39 Find an Equation for each of the eight regions in the Venn Diagram Write the equations in the form ? ?2+ ? ?2= ?2 The centre is on the line y=2x Touches the x-axis The radius is 5
Notes: The general equation of a circle Circle equation (? 5)2+(? 4)2= 65 Multiply out and make the equation = 0 [ ?2 10? + 25 ] + [ ?2 8? + 16 ] = 65 ?2+ ?2 10? 8? 24 = 0 This is called the General equation of the circle What links the two forms of the equations with centre (a, b) and radius r ? ??+ ?? ??? ?? ?? = 0 (? ?)?+(? ?)?= ?? -a a2 + b2 - r2 -2b -2a -b r2
WB40ab Find the centre and radius of each circle a) ?2+ ?2+ 4? 12? + 27 = 0 Rearrange to ?2+ 4? + 16 + ?2 12? + 36 25 = 0 Then to (? + 2)2+(? 6)2= 25 Centre (-2, 6) and radius 5 b) ?2+ ?2 14? + 2? + 34 = 0 Rearrange to ?2 14? + 49 + ?2+ 2? + 1 16 = 0 Then to (? 7)2+(? + 1)2= 16 Centre (7, -1) and radius 4
WB40cd Find the centre and radius of each circle c) ?2 6? + ?2 4? + 12 = 0 Rearrange to ?2 6? + 9 + ?2 4? + 4 1 = 0 Then to (? 3)2+(? 2)2= 1 Centre (3, 2) and radius 1 d) ?2+ ?2+ 6? + 8? + 16 = 0 Rearrange to ?2+ 6? + 9 + ?2+ 8? + 16 9 = 0 Then to (? + 3)2+(? + 4)2= 9 Centre (-3, -4) and radius 3
WB41 Intersections Find where the circle with the equation (? 5)2+(? 4)2= 65meets the x-axis ? 52+ 0 42= 65 ? 52= 49 ? 5 = 7 In the places where the equation meets the x-axis, the y-coordinate is 0 Sub this into the equation ? = 12 ?? ? = 2 So the coordinates are: 12,0 ??? ( 2,0)
WB42a Intersections Find the coordinates where the line y = x + 5 meets the circle x2 + (y 2)2 = 29. This is effectively solving simultaneous equations, ? = ? + 5 x = -5 x = 2 ? = 5 + 5 ? = 2 + 5 ?2+ ? + 5 22= 29 ? = 0 ? = 7 ?2+ ?2+ 6? + 9 = 29 2?2+ 6? 20 = 0 (2,7) ?2+ 3? 10 = 0 (0, 2) ? + 5 ? 2 = 0 (-5,0) ? = 5 ?? ? = 2 x2 + (y 2)2 = 29 y = x + 5 Sub these into the linear equation to find the y-coordinates
WB43 Intersections Start in the same way as the last question, by replacing y with x 7 in the circle equation Show that the line y = x 7 does not touch the circle (x + 2)2 + y2 = 33 (x + 2)2 + y2 = 33 (? + 2)2 + (? 7)2 = 33 y = x - 7 ?2+ 4? + 4 + ?2 14? + 49 = 33 2?2 10? + 20 = 0 ?2 5? + 10 = 0 ? =5 15 2 As we cannot square root a negative number, this equation is unsolvable The geometrical implication is that the lines do not meet!
KUS objectives BAT Find and use equations of circles BAT find the centre and radius of a circle from its equation self-assess One thing learned is One thing to improve is
