Circle Geometry Equations KUS objectives BAT Find and

Circle Geometry Equations • KUS objectives BAT Find and use the general equations of circles BAT solve geometry problems using your knowledge of circles Starter: (next page)

Challenge: Change one aspect so that the circles touch Geogebra activity

Example 1: The general equation of a circle Circle equation Multiply out and make the equation = 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 ? -a -b r 2 -2 a -2 b a 2 + b 2 - r 2

WB 12 Find the centre and radius of the circle with equation Rearrange to Centre (-2, 6) and radius 5

In the places where the equation meets the x-axis, the y-coordinate is 0 Sub this into the equation… So the coordinates are:

WB 14 a Intersections Find the coordinates where the line y = x + 5 meets the circle x 2 + (y – 2)2 = 29. This is effectively solving simultaneous equations, where one is a quadratic (although actually it is a circle) You can solve by substitution. Replace the y in the circle equation with x + 5 since we are told these are equivalent… We now know the x-coordinates where the lines meets are -5 and 2 Sub these into the linear equation to find the y-coordinates…

WB 14 b Intersections x = -5 (2, 7) x=2 (-5, 0) y = x + 5 x 2 + (y – 2)2 = 29

WB 15 Intersections Show that the line y = x – 7 does not touch the circle (x + 2)2 + y 2 = 33 Start in the same way as the last question, by replacing y with x – 7 in the circle equation… y = x - 7 As we cannot square root a negative number, this equation is unsolvable (x + 2)2 + y 2 = 33 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 –

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