The equation of a circle

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Presentation transcript:

The equation of a circle Consider a circle with centre (0,0) and radius r Consider some point (x,y) on the circumference Using Pythagoras: The same must always be true, provided (x,y) is on the circumference The function is a circle with centre (0,0) and radius r Complete the table Radius 3 10 Equation

What is the equation of the circle? The function is a circle with centre (0,0) and radius r What is the equation of the circle?

Show that the coordinates (6,7) and (-2,9) lie on the circle Points on the circumference of a circle Eg the graph shows the circle Eg the graph shows the circle (-2,9) (6,7) P(4,a) Q(b,-2) Show that the coordinates (6,7) and (-2,9) lie on the circle Find the values of a and b

1) 2) 3) 4) Find the exact value of the letters (-3,c) (2,a) (b,-3)

1) 2) 3) 4) Find the exact value of the letters (-3,c) (2,a) (b,-3)

Do these coordinates lie on, in or out of the circle? (3,8) (-5,7) (6,-6) (-8,-3)

Do these coordinates lie on, in or out of the circle? (3,8) (-5,7) (6,-6) (-8,-3) (-5,7) (3,8) (6,-6) (-8,-3)

Each of these circles have some integer coordinates on their circumference. How many? Why? 1) 2) 3) 4)

Each of these circles have some integer coordinates on their circumference. How many? Why? 1) 2) 3) 4) (6,8) (-6,8) (6,-8) (-6,-8) (8,6) (8,-6) (-8,6) (-8,-6) (4,4) (4,-4) (-4,4) (-4,-4) (3,11) (9,7) (11,3) (7,9) (-3,11) (-9,7) (-11,3) (-7,9) (3,-11) (9,-7) (11,-3) (7,-9) (-3,-11) (-9,-7) (-11,-3) (-7,-9) (1,7) (-1,7) (1,-7) (-1,-7) (5,5) (5,-5) (-5,5) (-5,-5) (7,1) (-7,1) (7,-1) (-7,-1)

Eg write down the equation of this circle Problem solving with circles The function is a circle with centre (0,0) and radius r Eg write down the equation of this circle Eg find the equation of this circle (-2,6) 4

Problem solving with circles The function is a circle with centre (0,0) and radius r 1. A circle has centre (0,0) and passes through the point (5,12). Find the equation of the circle and state its radius (5,12) 2. A circle has centre (0,0) and passes through the point (-3,7). Find the equation of the circle and state its radius (-3,7)

Problem solving with circles The function is a circle with centre (0,0) and radius r 1. A circle has centre (0,0) and passes through the point (5,12). Find the equation of the circle and state its radius (5,12) 2. A circle has centre (0,0) and passes through the point (-3,7). Find the equation of the circle and state its radius (-3,7)

3. The diagram shows a circle, centre (0,0). The points (4,-5) and (c,3) lie on the circle. Find the exact value of c (c,3) (4,-5) 4. The diagram shows the graph of Find the exact values of y when x = ½

3. The diagram shows a circle, centre (0,0). The points (4,-5) and (c,3) lie on the circle. Find the exact value of c (c,3) (4,-5) 4. The diagram shows the graph of Find the exact values of y when x = ½

Tangents to circles A tangent to a circle is a line that touches the circle once. You have seen that a tangent at a point is perpendicular to a radius to the same point You have also seen that if a line has gradient m, then a perpendicular line has gradient (2,6) (-10,5) (-4,-6)

Find the equation of a tangent to the circle at (2,4) Finding equations of tangents to circles If a radius has gradient m, then a tangent has gradient Eg the graph shows the circle (2,4) Find the equation of a tangent to the circle at (2,4)

Finding equations of tangents to circles 1. The graph shows the circle Find the equation of a tangent to the circle at (3,9) (3,9)

Finding equations of tangents to circles 1. The graph shows the circle Find the equation of a tangent to the circle at (3,9) (3,9)

2. Here is a circle, centre O, and the tangent to the circle at the point P(4, 3) on the circle. Find an equation of the tangent at the point P

2. Here is a circle, centre O, and the tangent to the circle at the point P(4, 3) on the circle. Find an equation of the tangent at the point P

3. The graph shows the circle Find the value of a Find the equation of a tangent to the circle at P P(-3,a)

3. The graph shows the circle Find the value of a Find the equation of a tangent to the circle at P P(-3,a)

4. The line l is a tangent to the circle x2 + y2 = 40 at the point A 4. The line l is a tangent to the circle x2 + y2 = 40 at the point A. A is the point (2, 6). The line l crosses the x–axis at the point P. Work out the area of triangle OAP.

The line l crosses the x–axis at the point P. 4. The line l is a tangent to the circle x2 + y2 = 40 at the point A. A is the point (2, 6). The line l crosses the x–axis at the point P. Work out the area of triangle OAP. A(2,6) O P substitute y = 0 into to find P

The diagram shows the circle P O Q P lies on the circle and has x-coordinate 1. The tangent at P interests the x-axis at Q. Work out the coordinates of Q. Notice anything unusual? Investigate! y-coordinate of P is 3 substitute y = 0 into to find Q