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GCSEMathsAQAHigherFlashcards2. AlgebraEquation of a Circle

Equation of a Circle (AQA GCSE Maths)

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  • Write down the equation of a circle, centred at the origin, with a radius of r.

    The equation of a circle centred at the origin with a radius of r is:

    x squared plus y squared equals r squared

  • How do you find the radius of the circle given by x squared plus y squared equals 81?

    To find the radius of the circle given by x squared plus y squared equals 81 you need to square root 81.

    This is because r squared equals 81 from the equation x squared plus y squared equals r squared.

    This gives a radius of 9.

    You do not need plus-or-minus 9, as a radius is always positive.

  • Write down the equation of the circle shown.

    A circle with radius 5 on a grid.

    The equation is x squared plus y squared equals 25 as the radius is 5 so r equals 5 in x squared plus y squared equals r squared.

    Graph showing a circle centered at the origin with radius 5, plotted on a Cartesian coordinate system with x and y axes ranging from -7 to 7.

  • How would you find the diameter of the circle x squared plus y squared equals 100?

    To find the diameter of the circle x squared plus y squared equals 100, first compare it to the equation of a circle given by x squared plus y squared equals r squared.

    You can see that the radius, r, is 10, because r squared equals 100.

    The diameter is double the radius. This means the diameter is 20.

  • True or False?

    The radius of the circle x squared plus y squared equals 8 is square root of 8.

    True.

    The radius of the circle x squared plus y squared equals 8 is square root of 8.

    It is possible for a radius to be left as a square root.

    You can also use surd methods to simplify them, such as square root of 8 equals square root of 4 cross times 2 end root equals square root of 4 cross times square root of 2 equals 2 square root of 2.

  • True or False?

    The point open parentheses 8 comma space 6 close parentheses lies on the circumference of the circle x squared plus y squared equals 100.

    True.

    The point open parentheses 8 comma space 6 close parentheses lies on the circumference of the circle x squared plus y squared equals 100.

    You can show this by substituting in x equals 8 and y equals 6 into the left-hand side.

    This gives 8 squared plus 6 squared equals 64 plus 36 equals 100.

    This matches the right-hand side, so the equation is satisfied (the point lies on the circle).

  • How can you find out if the point open parentheses 4 comma space 3 close parentheses lies inside or outside the circle x squared plus y squared equals 36?

    To find out if the point open parentheses 4 comma space 3 close parentheses lies inside or outside the circle x squared plus y squared equals 36, first substitute x equals 4 and y equals 3 into the left-hand side of the equation.

    This gives 4 squared plus 3 squared equals 16 plus 9 equals 25.

    Compare this number to the right-hand side of the equation. If it is less than the right-hand side, the point lies inside the circle.

    25 less than 36 so the point lies inside the circle.

  • True or False?

    The y-intercepts of the circle x squared plus y squared minus 2 equals 0 are plus-or-minus 2.

    False.

    The equation rearranges to x squared plus y squared equals 2.

    This means the radius is square root of 2 (because x squared plus y squared equals r squared and r squared equals 2).

    The y-intercepts of a circle centred at the origin are plus-or-minus r (plus or minus the radius).

    Therefore the y-intercepts are plus-or-minus square root of 2.

  • True or False?

    The equation x squared plus y squared plus 1 equals 0 represents a circle.

    False.

    The equation rearranges to x squared plus y squared equals negative 1, which has a negative right-hand side.

    The equation of a circle is x squared plus y squared equals r squared, so the right-hand side must always be positive.

  • Explain why the line x equals 12 never intersects the circle x squared plus y squared equals 100.

    The circle x squared plus y squared equals 100 is centred at the origin with a radius of 10.

    The line x equals 12 is a vertical line through 12 on the x-axis.

    This will never intersect the circle, as the circle lies between plus-or-minus 10 on the x-axis.

  • True or False?

    A radius which passes through a point P on a circle, is perpendicular to the tangent to the circle which passes through point P.

    True.

    A radius which passes through a point P on a circle, is perpendicular to the tangent to the circle which passes through point P.

  • If the gradient of the radius passing through point P on a circle is known, how can you find the gradient of the tangent to the circle at point P?

    If the gradient of the radius passing through point P on a circle is known, the gradient of the tangent to the circle at point P will be its negative reciprocal.

    This is because a radius and a tangent which meet at a point, will meet at right angles.

  • Given a point open parentheses a comma space b close parentheses on the circumference of a circle with centre open parentheses 0 comma space 0 close parentheses, outline how you would find the equation of the tangent passing through open parentheses a comma space b close parentheses.

    Given a point open parentheses a comma space b close parentheses on the circumference of a circle with centre open parentheses 0 comma space 0 close parentheses, outline how to find the equation of the tangent passing through open parentheses a comma space b close parentheses:

    • Find the gradient of the radius: fraction numerator b minus 0 over denominator a minus 0 end fraction equals b over a.

    • Use the fact that the tangent is perpendicular to this, so has gradient negative a over b.

    • The equation of the tangent will be in the form y equals m x plus c.

    • Substitute the gradient negative a over b, and the point open parentheses a comma space b close parentheses into y equals m x plus c to find c

Previous:Graphs of FunctionsNext:Estimating Gradients & Areas under Graphs

    1. Topic 1

      Number Operations

    2. Topic 2

      Prime Factors, HCF & LCM

    3. Topic 3

      Powers, Roots & Standard Form

    4. Topic 4

      Fractions

    5. Topic 5

      Percentages

    6. Topic 6

      Simple & Compound Interest, Growth & Decay

    7. Topic 7

      Fractions, Decimals & Percentages

    8. Topic 8

      Rounding, Estimation & Bounds

    9. Topic 9

      Surds

    10. Topic 10

      Using a Calculator

    1. Topic 1

      Introduction to Algebra

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Linear Equations

    10. Topic 10

      Quadratic Equations

    11. Topic 11

      Simultaneous Equations

    12. Topic 12

      Iteration

    13. Topic 13

      Forming & Solving Equations

    14. Topic 14

      Functions

    15. Topic 15

      Coordinate Geometry

    16. Topic 16

      Linear Graphs

    17. Topic 17

      Graphs of Functions

    18. Topic 18

      Equation of a Circle

    19. Topic 19

      Estimating Gradients & Areas under Graphs

    20. Topic 20

      Real-Life Graphs

    21. Topic 21

      Solving Inequalities

    22. Topic 22

      Graphing Inequalities

    23. Topic 23

      Transformations of Graphs

    24. Topic 24

      Sequences

    1. Topic 1

      Ratios

    2. Topic 2

      Problem Solving with Ratios

    3. Topic 3

      Direct & Inverse Proportion

    4. Topic 4

      Standard & Compound Units

    5. Topic 5

      Exchange Rates & Best Buys

    1. Topic 1

      Symmetry & Shapes

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing, Constructions & Loci

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    13. Topic 13

      Vectors

    14. Topic 14

      Transformations

    1. Topic 1

      Introduction to Probability

    2. Topic 2

      Simple Probability Diagrams

    3. Topic 3

      Tree Diagrams

    4. Topic 4

      Combined & Conditional Probability

    1. Topic 1

      Averages, Ranges & Data

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency & Box Plots

    5. Topic 5

      Scatter Graphs & Correlation