1. S TARTER Can you write an equation that will produce a circle radius 6? Can you write an equation that will produce a circle radius 2? What is the centre of this circle? Write an equation that gives a circle with centre (3,9) Write down the radius of this circle

2. S TARTER Can you suggest an equation for this circle?

3. S TARTER Can you suggest an equation for this circle?

4. S TARTER Can you suggest an equation for a circle that is only in the first quadrant?

5. S TARTER Can you suggest an equation for a circle that touches the x and y axes?

6. C IRCLES 2

7. F IND THE EQUATION OF THE TANGENT TO THE CIRCLE AT THE POINT (6,2) (6,2) (4,1)

8. F IND THE EQUATION OF THE NORMAL TO THE CIRCLE AT THE POINT (6,2) (6,2) (4,1) normal

9. F INDING THE E QUATION OF A TANGENT Step 1 – Find the centre of the circle Rearrange the circle formula if necessary Step 2 – Find the gradient of the radius Use the centre and point on the edge Step 3 – Find the gradient of the tangent Use the fact that m r x m t = -1 Step 4 – Find the equation of the tangent using the gradient and the point given Use y – y 1 = m ( x - x 1 ) with m t and the point

10. T ANGENTS AND N ORMALS E XERCISE Equation of normal isEquation of tangent is

11. F IND THE I NTERSECTION OF and

12. F IND THE I NTERSECTION OF and

13. F IND THE I NTERSECTION OF and

14. F INDING POINTS OF INTERSECTION Step 1 – Rearrange the straight line equation Use either or which ever is easiest Step 2 – Substitute into the circle equation Swap the rearrangement for x or y Step 3 – Expand any brackets and simplify Remove brackets and collect like terms Step 4 – Solve the quadratic equation Use either factorisation or the formula Step 5 – Substitute back in to find coordinates You must find the full coordinate pair (x, y)

15. F INDING POINTS OF INTERSECTION You may be asked to find the points of intersection between a circle and a line. Remember 3 things that could happen For points of intersections always think Simultaneous Equations! To say how many times it will intersect think Discriminant!

16. F INDING POINTS OF INTERSECTION Find the points of intersection of the circle and the line

17. T ANGENTS TO C IRCLES – C HALLENGE ! The line with equation is a tangent to the circle Find the possible values of k.

18. If the line is a tangent the simultaneous equation will have 0 / 1 / 2 solutions? Substitute into the circle equation Multiply out To check for one solution, use the discriminant,

19. Substitute into the discriminant Multiply it all out Collect like terms together Divide everything by -4 Factorise So (-2,3)

20. E XAM Q UESTION A circle with centre C has equation (a) By completing the square, express this equation in the form (b) Write down (i) the coordinates of C (ii) the radius of the circle (c) Show that the circle does not intersect the x-axis (d) The line with equation x + y = 4 intersects the circle at the points P and Q ( i) Show that the x-coordinates of P and Q satisfy the equation (ii) Given that P has coordinates (2,2), find the coordinates of Q (iii) Hence find the coordinates of the midpoint of PQ

21. E XAM Q UESTION A circle with centre C has equation (a) By completing the square, express this equation in the form (b) Write down (i) the coordinates of C (ii) the radius of the circle

22. E XAM Q UESTION (c) Show that the circle does not intersect the x-axis (d) The line with equation x + y = 4 intersects the circle at the points P and Q ( i) Show that the x-coordinates of P and Q satisfy the equation (ii) Given that P has coordinates (2,2), find the coordinates of Q (iii) Hence find the coordinates of the midpoint of PQ

23. E XAM Q UESTION (c) Show that the circle does not intersect the x-axis (d) The line with equation x + y = 4 intersects the circle at the points P and Q ( i) Show that the x-coordinates of P and Q satisfy the equation (ii) Given that P has coordinates (2,2), find the coordinates of Q (iii) Hence find the coordinates of the midpoint of PQ