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The points X and Y have coordinates (0,3) and (6,11) respectively. XY is a chord of a circle C with centre Z.

a) Find the gradient of XY.

The point M is the midpoint of XY

b) find an equation for the line which passes through Z and M.

Given that the y coordinate of Z is 10

c) find the x coordinate of Z

d) find the equation of the circle C, giving your answer in the form x^2+y^2+ax+by+c=0 where a, b and c are constants.

a) Find the gradient of XY.

The point M is the midpoint of XY

b) find an equation for the line which passes through Z and M.

Given that the y coordinate of Z is 10

c) find the x coordinate of Z

d) find the equation of the circle C, giving your answer in the form x^2+y^2+ax+by+c=0 where a, b and c are constants.

a) Change in y divided by the change in x

(11-3)/(6-0) = 4/3

b) Find the midpoint M(3,7)

The line ZM is perpendicular to the line XY, meaning that the gradients of the two lines multiply to make -1, which makes the gradient of ZM -3/4.

y = -3/4x + c

'plug' the coordinates of M into this.

7 = -3/4(3) + c

c = 37/4

y = -3/4 x+ 37/4

c) 10 = -3/4x +37/4

3/4 = -3/4x

x = -1

d) Use Pythagoras' theorem to work out the lengths of ZM and YM

ZM:

4^2 + 3^2 = 25 (5^2)

ZM=5

YM:

3^2 + 4^2 =35 (5^2)

YM=5

ZY (radius):

5^2 +5^2 =50

the radius is the square root of 50, but the equation of a circle wants the radius squared, which is 50

(x+1)^2 + (y-10)^2 =50

x^2 + 2x + 1 + y^2 -20y +100 = 50

x^2 + y^2 + 2x - 20y +51 = 0

(11-3)/(6-0) = 4/3

b) Find the midpoint M(3,7)

The line ZM is perpendicular to the line XY, meaning that the gradients of the two lines multiply to make -1, which makes the gradient of ZM -3/4.

y = -3/4x + c

'plug' the coordinates of M into this.

7 = -3/4(3) + c

c = 37/4

y = -3/4 x+ 37/4

c) 10 = -3/4x +37/4

3/4 = -3/4x

x = -1

d) Use Pythagoras' theorem to work out the lengths of ZM and YM

ZM:

4^2 + 3^2 = 25 (5^2)

ZM=5

YM:

3^2 + 4^2 =35 (5^2)

YM=5

ZY (radius):

5^2 +5^2 =50

the radius is the square root of 50, but the equation of a circle wants the radius squared, which is 50

(x+1)^2 + (y-10)^2 =50

x^2 + 2x + 1 + y^2 -20y +100 = 50

x^2 + y^2 + 2x - 20y +51 = 0

The demand function facing a monopolist selling two products are given as

Q1=40-2p-p2

Q2=35-p1-p2

Solve for p1 and p2 in terms of Q1 and Q2 in the demand functions to form the average revenue function

Q1=40-2p-p2

Q2=35-p1-p2

Solve for p1 and p2 in terms of Q1 and Q2 in the demand functions to form the average revenue function

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