CIE/GCE/AS/Math/P1/20/Mar/12/Q#12
Question
A diameter of a circle C1 has end-points at (−3, −5) and (7, 3).
a) Find an equation of the circle C1.
8
The circle C1 is translated by ( ) to give circle C2, as shown in the diagram.
4
b) Find an equation of the circle C2.
The two circles intersect at points R and S.
c) Show that the equation of the line RS is y = −2x + 13.
d) Hence show that the x-coordinates of R and S satisfy the equation
5x2 − 60x + 159 = 0
Solution
a)
, CIE/GCE/AS/Math/P1/20/Mar/12/Q#12
We are given that points at (−3, −5) and (7, 3) makes end-points of the diameter of a
circle C1.
If coordinates of end points of diameter, 𝐴 and 𝐵 are given;
✓ Find the coordinates of mid-point of 𝐴𝐵, this gives coordinates of center of the
circle 𝐶(𝑎, 𝑏)
✓ Find the distance 𝐶𝐴 or 𝐶𝐵 , this gives the radius of the circle 𝑟
✓ Use the equation of the circle (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
Therefore, first we find the midpoint of given end-points A(−3, −5) and B(7, 3) of
diameter of the circle C1.
To find the mid-point of a line we must have the coordinates of the end-points of the
line.
Expressions for coordinates of mid-point of a line joining points 𝑃1 (𝑥1 , 𝑦1 ) and𝑃2 (𝑥2 , 𝑦2 );
1
x-coordinate of mid-point 𝑀(𝑥, 𝑦) of the line= 𝑥𝑀 = (𝑥1 + 𝑥2 )
2
1
y-coordinate of mid-point 𝑀(𝑥, 𝑦) of the line= 𝑦𝑀 = (𝑦1 + 𝑦2 )
2
Hence;
1 1 1
x-coordinate of mid-point 𝐶(𝑥, 𝑦) of the line= 𝑥𝐶 = 2 (𝑥𝐴 + 𝑥𝐵 ) = 2 (−3 + 7) = 2 (4) = 2
1 1 1
y-coordinate of mid-point 𝐶(𝑥, 𝑦) of the line= 𝑦𝐶 = 2 (𝑦𝐴 + 𝑦𝐵 ) = 2 (−5 + 3) = 2 (−2) = −1
Hence, the point 𝐶(2, −1) is the center of the circle 𝐶1 .
Next, we find the radius of the circle 𝐶1 by finding either distance CA or CB.
We choose to find CB.
Expression to find distance between two given points 𝐴(𝑥𝐴 , 𝑦𝐴 ) and 𝐵(𝑥𝐵 , 𝑦𝐵 )is:
Question
A diameter of a circle C1 has end-points at (−3, −5) and (7, 3).
a) Find an equation of the circle C1.
8
The circle C1 is translated by ( ) to give circle C2, as shown in the diagram.
4
b) Find an equation of the circle C2.
The two circles intersect at points R and S.
c) Show that the equation of the line RS is y = −2x + 13.
d) Hence show that the x-coordinates of R and S satisfy the equation
5x2 − 60x + 159 = 0
Solution
a)
, CIE/GCE/AS/Math/P1/20/Mar/12/Q#12
We are given that points at (−3, −5) and (7, 3) makes end-points of the diameter of a
circle C1.
If coordinates of end points of diameter, 𝐴 and 𝐵 are given;
✓ Find the coordinates of mid-point of 𝐴𝐵, this gives coordinates of center of the
circle 𝐶(𝑎, 𝑏)
✓ Find the distance 𝐶𝐴 or 𝐶𝐵 , this gives the radius of the circle 𝑟
✓ Use the equation of the circle (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
Therefore, first we find the midpoint of given end-points A(−3, −5) and B(7, 3) of
diameter of the circle C1.
To find the mid-point of a line we must have the coordinates of the end-points of the
line.
Expressions for coordinates of mid-point of a line joining points 𝑃1 (𝑥1 , 𝑦1 ) and𝑃2 (𝑥2 , 𝑦2 );
1
x-coordinate of mid-point 𝑀(𝑥, 𝑦) of the line= 𝑥𝑀 = (𝑥1 + 𝑥2 )
2
1
y-coordinate of mid-point 𝑀(𝑥, 𝑦) of the line= 𝑦𝑀 = (𝑦1 + 𝑦2 )
2
Hence;
1 1 1
x-coordinate of mid-point 𝐶(𝑥, 𝑦) of the line= 𝑥𝐶 = 2 (𝑥𝐴 + 𝑥𝐵 ) = 2 (−3 + 7) = 2 (4) = 2
1 1 1
y-coordinate of mid-point 𝐶(𝑥, 𝑦) of the line= 𝑦𝐶 = 2 (𝑦𝐴 + 𝑦𝐵 ) = 2 (−5 + 3) = 2 (−2) = −1
Hence, the point 𝐶(2, −1) is the center of the circle 𝐶1 .
Next, we find the radius of the circle 𝐶1 by finding either distance CA or CB.
We choose to find CB.
Expression to find distance between two given points 𝐴(𝑥𝐴 , 𝑦𝐴 ) and 𝐵(𝑥𝐵 , 𝑦𝐵 )is:
