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.

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Presentation transcript:

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

S TARTER Can you suggest an equation for this circle?

S TARTER Can you suggest an equation for this circle?

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

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

C IRCLES 2

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

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

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

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

F IND THE I NTERSECTION OF and

F IND THE I NTERSECTION OF and

F IND THE I NTERSECTION OF and

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)

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!

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

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

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,

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

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

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

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

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