Circle Exam Questions

Questions

• 1. A circle as the centre (5,8) and has a radius of 5.
• a). What is the equation of the circle
• b). verify point (2,12) lies within the circle
• c). Find the equation of the tangent that scrapes (2,12) in the form ay-bx-c=0 where a b and c are integers
• d). (7,12) is the midpoint of chord PQ. Find the distance of the midpoint from the origin
• e). Find the area of the triangle PCQ (C=centre)
• 2. A circle pass through the points (1,3) , (5,7) , (2,9). Find the equation of the cirlce.
• 3. y=mx is a tangent to x^2+y^2-10y+16=0. Find 2 possible values for m.

Answers 1

• a). (x-5)^2+(y-8)^2=25
• b).(2-5)^2+(12-8)^2 , (-3)^2 +(4)^2 = 25 , so it lies within the circle
• c). Change in y / change in x from centre to (2,12) to get y=-4/3x. Now the negative reciprocal of this is 3/4 (3/4 x -4/3 = -1). Now sub in (2,12) to y=3/4x + c , and find that c=21/2. Now rearrange to get 4y-3x-42 = 0.
• d). Change is y=2 and change in x=4, and pythag of this gives 2(root5)
• e). root5 x 2(root5) = 10.

Answers 2

• Sub coordinates in to get: (1-a)^2 + (3-b)^2 = r^2 , (5-a)^2 + (7-b)^2 = r^2 , (2-a)^2 + (9-b)^2 = r^2
• 1st and 2nd expanded ; 1-2a+a^2 +9-6b+b^2 = 25-10a+a^2+49-14b+b^2 , 10-2a-6b=74-10a-14b , 8a+8b=64 , a+b=8
• 1st and 3rd expanded ; 1-2a+a^2+9-6b+b^2=4-4a+a^2+81-18b+b^2 , 10-2a-6b=85-4a-18b , 2a+12b=75
• Solve simultaneous equation : a+b=8 // 2a+12b=75 to get a=2.1 and b=5.9 , so (x-2.1)^2+(y-5.9)^2 = r^2
• Finally sub in a coordinate to this equation, so for example: (1-2.1)^2+(3-5.9)^2 = r^2, and solve for r^2 which is 9.62
• Now we have the equation of the circle, which is (x-2.1)^2+(y-5.9)^2=9.62

Answer 3

• 3. y=mx is a tangent to x^2+y^2-10y+16=0. Find 2 possible values for m.
• Sub in mx to y, so: x^2 + (mx)^2 - 10(mx) +16 = 0 , x^2 + m^2x^2 - 10mx +16 = 0 , (1+m^2)x^2-10mx+16=0.
• Now as we know the tangent meets the circle at a single point, we can use the discriminant which must equal 0. So: (-10m)^2 - 4(2+m^2)(16) , 100m^2 - 64 - 64m^2 , 36m^2 = 64 , m^2 = 64/36 , m= +/- 4/3.