Equation of a Circle with Centre Not at Origin
The equation of a circle is explored when the centre is not at the origin, with examples and problems to solve intersections, find centre and radius, related points, and shifted centre. Visual aids and quickfire questions enhance understanding of circle equations. Circumference points and y-axis tangent circle are analyzed, along with determining point coordinates.
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Equation of a Circle Write the equation of a circle where the centre is not the origin Justify if points are on the circumference Solve problems with intersections
GCSE Skills Recap ?(15,5) What is the: a) Centre of ? and ?? ?,?? ? ?(3,0) ? b) The length ??? ? = ???+ ??? = ???+ ?? = ?? ?
GCSE Skills Recap The equation of this circle is: ? x2 + y2 = 25 ? 5 ? 5 -5 The equation of a circle with centre at the origin and radius r is: ?2+ ?2= ?2 -5
Equation of a Circle How could we show how ?, ? and ? are related? (Hint: draw a right-angled triangle inside your circle, with one vertex at the origin and another at the circumference) ? ?,? ? ? ? ?2+ ?2= ?2 ? ? ? ?
Equation of a Circle Find an equation for the length of the hypotenuse: ? (?,?) ? (3,2) ?
Equation of a Circle Now suppose we shift the circle so it s now centred at (a, b). What s the equation now? (Hint: What would the sides of this right-angled triangle be now?) The equation of a circle with radius ? and centre (?,?) is: ? ?2+ ? ?2= ?2 ? ? (?,?) ?
Quickfire Questions Centre Radius Equation x2 + y2 = 25 ? ? (0,0) 5 (x-1)2 + (y-2)2 = 36 (1,2) 6 (-3,5) ? ? ? ? ? 1 ? ? ? ? ? (x+3)2 + (y-5)2 = 1 (x+5)2 + (y-2)2 = 49 (-5,2) 7 (x+6)2 + (y+7)2 = 16 (-6,-7) 4 (x-1)2 + (y+1)2 = 3 (1,-1) 3 ?2+ ? 32= 8 (0,3) 2 2 The equation of a circle with radius ? and centre (?,?) is: ? ?2+ ? ?2= ?2 Help!
Equation of a Circle The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in the diagram. Write down an equation for the circle C. ? The equation of a circle with radius ? and centre (?,?) is: ? ?2+ ? ?2= ?2
Points on the Circumference A circle has centre (3,4) and has the radius 5. The circle intercepts the ?-axis at the origin and the point ? and the ?-axis at the origin and the point ?. Determine the coordinates of these points. ? Equation is: ? 32+ ? 42= 25 ? ? When ? = 0: (3,4) ? 32+ 16 = 25 ? 32= 9 ? 3 = 3 ? = 0 ?? 6 ? 6,0 When ? = 0: 9 + ? 42= 25 ? = 0 ?? 8 ? ? ? ? ? 0,8
Your Turn Does the circle with equation ?2+ ? 12= 16 pass through the point (2,5)? A circle has centre 4,7 , radius 10. Find the coordinates of the points on the circle where ? = 13. ? 42+ ? 72= 100 ?2+ ? 12= 22+ 42= 20 When ? = 13: ? 42+ 13 72= 100 ? 42+ 36 = 100 ? 42= 64 ? 4 = 8 ? = 4 ?? 12 4,13 , 12,13 Since 20 16 it is not on the circle. ? ?
The diagram shows a sketch of the circle ? 72+ ? 42= 9. The line ? = 6 intersects the circle at the points ? and ?. Show that ?? = 2 5 ? 72+ 6 42= 9 ? 72+ 4 = 9 ? 7 = 5 ? = 7 5 ? ? = 7 5,6 ? = (7 + 5,6) ?? = 7 + 5 7 5 = 2 5
Exercise 1 1 Write down the equation of each of these circles: ??+ ? ??= ? ? ??+ ? + ??= ?? ? + ??+ ? ??= ? ? ??+ ? ???= ??? ? ? ? ? (a) Centre (0,3) radius 2. (b) Centre (1, -5) radius 4 (c) Centre (-3, 4) radius 7 (d) Centre (8, 15) radius 17 Does the circle pass through the origin? Show working to support your answer. ? Yes as ??+ ???= ??? Write down the centre and radius of each of these circles. 2 (a) ?2+ ?2= 36 (b) ? 32+ ? 42= 100 (a) ? + 52+ ?2= 3 Centre (0,0) Radius = 6 Centre (3,4) Radius = 10 Centre (-5,0) Radius = ? ? ? ? Determine whether the point ? lies on each of these circles. 3 ? ? 1,1 on ?2+ ? + 22= 10Yes. ?(4,7) on ? 12+ ? 22= 9x No. (a) (b) ?
Exercise 1 [AQA] A circle has equation ? 52+ ? 42= 100 Show that the point 13, 2 lies on the circle. ?? ??+ ? ??= ??? 4 ? [AQA] The point 13, 2 lies on the circle 5 ? ?2+ ? 42= 100 Work out the two possible values of ?. ? = ? ?? ?? ? [Jan 2013 Paper 2] Match each statement with an equation. You will not use all the equations. 6 ?
Exercise 1 [Jan 2012 Paper 2] A circle has the equation ?2+ ?2= 36. Work out its circumference. Radius = 6 Circumference = ??? ? 7 [June 2013 Paper 2] The circle ?2+ ?2= 25 touches each side of the square as shown. Work out the total shaded area. 8 Area = ??? ? ?? = ??? ??? ?
Exercise 1 Circle A has equation: ?2+ ?2= 16 Circle ? has equation ? + 62+ ? 82= 25 [AQA] 9 (a) Work out the distance between the centres of the circles. (b) Circle the correct statement: The circles overlap The circles touch Distance between (?,?) and ?,? is ?? ? The circles do not overlap The circles do not overlap as the sum of the radii is less than the difference between the centres: ? + ? = ? < ?? ? Determine the coordinates of the points where the circle with equation: ? + 52+ ? 122= 169 10 intercepts both axes. When ? = ?, ? = ? ?? ?? ?,? , ?,?? When ? = ?,? = ? ?? ?? ?,? , ??,? ? 11 Determine the points ? and ? where the circle with equation: ? 42+ ? + 32= 100 intersects the line with equation ? = 12. Hence determine the length ??. ? 12, 9 ?(12,3) ?? = ?? ?
