2. Distance Between Two Points NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME The Distance Formula d = ( y 2 – y 1 ) ² + ( x 2 – x 1 ) ² √ B ( x 2 , y 2 ) A ( x 1 , y 1 ) y 2 – y 1 x 2 – x 1 Example Calculate the distance between (-2,9) and (4,-3). d = + 6 ² √ 12 ² = 180 √ = 5 √ 6 Where required, write answers as a surd in its simplest form. REMEMBER
3. Points on a Circle NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME Example Plot the following points and find a rule connecting x and y . x y ( 5 , 0 ) ( 4 , 3 ) ( 3 , 4 ) ( 0 , 5 ) (-3 , 4 ) (-4 , 3 ) (-5 , 0 ) (-4 ,-3 ) (-3 ,-4 ) ( 0 ,-5 ) ( 3 ,-4 ) ( 4 ,-3 ) All points lie on a circle with radius 5 units and centre at the origin. x ² + y ² = 25 x ² + y ² = r ² For any point on the circle, For any radius... NOTICE
4. The Equation of a Circle with centre at the Origin NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME x ² + y ² = r ² For any circle with radius r and centre the origin, x y The ‘Origin’ is the point (0,0) origin Example Show that the point ( - 3 , ) lies on the circle with equation 7 x ² + y ² = 16 x ² + y ² = ( -3 ) ² + ( ) ² 7 = 9 + 7 = 16 Substitute point into equation: The point lies on the circle. LEARN THIS
5. The Equation of a Circle with centre ( a , b ) NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME ( x – a ) ² + ( y – b ) ² = r ² For any circle with radius r and centre at the point ( a , b ) ... Not all circles are centered at the origin. x y ( a , b ) r Example Write the equation of the circle with centre ( 3 , -5 ) and radius 2 3 . ( x – a ) ² + ( y – b ) ² = r ² ( x – 3 ) ² + ( y – ( -5 ) ) ² = ( ) ² 2 3 ( x – 3 ) ² + ( y + 5 ) ² = 12 LEARN THIS
6. NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME The General Equation of a Circle ( x + g ) 2 + ( y + f ) 2 = r 2 ( x 2 + 2 g x + g 2 ) + ( y 2 + 2 fy + f 2 ) = r 2 x 2 + y 2 + 2 g x + 2 f y + g 2 + f 2 – r 2 = 0 x 2 + y 2 + 2 g x + 2 f y + c = 0 c = g 2 + f 2 – r 2 r 2 = g 2 + f 2 – c r = g 2 + f 2 – c Try expanding the equation of a circle with centre ( - g , - f ) . General Equation of a Circle with center ( - g , - f ) and radius r = g 2 + f 2 – c this is just a number... LEARN
7. NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME Circles and Straight Lines A line and a circle can have two, one or no points of intersection. r A line which intersects a circle at only one point is at 90° to the radius and is is called a tangent . two points of intersection one point of intersection no points of intersection REMEMBER
8. NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME Intersection of a Line and a Circle Example Find the intersection of the circle and the line 2 x – y = 0 x 2 + ( 2 x ) 2 = 45 x 2 + 4 x 2 = 45 5 x 2 = 45 x 2 = 9 x = 3 or -3 y = 2 x x 2 + y 2 = 45 Substitute into y = 2 x : How to find the points of intersection between a line and a circle: • rearrange the equation of the line into the form y = m x + c • substitute y = m x + c into the equation of the circle • solve the quadratic for x and substitute into m x + c to find y y = 6 or -6 Points of intersection are ( 3,6 ) and ( -3,-6 ) .
9. NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME Intersection of a Line and a Circle (continued) Example 2 Find where the line 2 x – y + 8 = 0 intersects the circle x 2 + y 2 + 4 x + 2 y – 20 = 0 x y x 2 + ( 2 x + 8 ) 2 + 4 x + 2 ( 2 x + 8 ) – 20 = 0 x 2 + 4 x 2 + 32 x + 64 + 4 x + 4 x + 16 – 20 = 0 5 x 2 + 40 x + 60 = 0 5 ( x 2 + 8 x + 12 ) = 0 5 ( x + 2 )( x + 6 ) = 0 x = -2 or -6 Substituting into y = 2 x + 8 points of intersection as ( -2,4 ) and ( -6,-4 ) . Factorise and solve
10. NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME The Discriminant and Tangents x = - b b 2 – ( 4 ac ) ± 2 a b 2 – ( 4 ac ) Discriminant The discriminant can be used to show that a line is a tangent: • substitute into the circle equation • rearrange to form a quadratic equation • evaluate the discriminant y = m x + c b 2 – ( 4 ac ) > 0 Two points of intersection b 2 – ( 4 ac ) = 0 The line is a tangent b 2 – ( 4 ac ) < 0 No points of intersection r REMEMBER
11. NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME Circles and Tangents Show that the line 3 x + y = -10 is a tangent to the circle x 2 + y 2 – 8 x + 4 y – 20 = 0 Example x 2 + (- 3 x – 10 ) 2 – 8 x + 4 (- 3 x – 10 ) – 20 = 0 x 2 + 9 x 2 + 60 x + 100 – 8 x – 12 x – 40 – 20 = 0 10 x 2 + 40 x + 40 = 0 b 2 – ( 4 ac ) = 40 2 – ( 4 × 10 × 40 ) = 0 = 1600 – 1600 The line is a tangent to the circle since b 2 – ( 4 ac ) = 0 x y
12. NOTE SLIDE Higher Maths 2 4 Circles UNIT OUTCOME Equation of Tangents To find the equation of a tangent to a circle: • Find the center of the circle and the point where the tangent intersects • Calculate the gradient of the radius using the gradient formula • Write down the gradient of the tangent • Substitute the gradient of the tangent and the point of intersection into y – b = m ( x – a ) REMEMBER Straight Line Equation y – b = m ( x – a ) m tangent = – 1 m radius x 2 – x 1 y 2 – y 1 m radius = r
