Mathematics Multiple Choice Questions on “Conic Sections – Circle”.
1. Find the equation of circle with center at origin and radius 5 units.
a) x2+y2=25
b) x2+y2=5
c) x2=25
d) y2=25
Answer: a
Clarification: Equation of circle with center at (a, b) and radius r units is
(x-a)2+(y-b)2=r2
So, equation of circle is (x-0)2+(y-0)2=52 => x2+y2=25.
2. Find the equation of circle with center at (2, 5) and radius 5 units.
a) x2+y2+4x-10y+4=0
b) x2+y2-4x-10y+4=0
c) x2+y2+4x+10y+4=0
d) x2+y2+4x-10y-4=0
Answer: b
Clarification: Equation of circle with center at (a, b) and radius r units is
(x-a)2+(y-b)2=r2
So, equation of circle is (x-2)2+(y-5)2=52 => x2+y2-4x-10y+4=0.
3. Find the center of the circle with equation x2+y2-4x-10y+4=0.
a) (-2, 5)
b) (-2, -5)
c) (2, -5)
d) (2, 5)
Answer: d
Clarification: Comparing the equation with general form x2+y2+2gx+2fy+c=0, we get
2g=-4 => g=-2
2f=-10 => f=-5
c=4
Center is at (-g, -f) i.e. (2, 5).
4. Find the radius of the circle with equation x2+y2-4x-10y+4=0.
a) 25 units
b) 20 units
c) 5 units
d) 10 units
Answer: c
Clarification: Comparing the equation with general form x2+y2+2gx+2fy+c=0, we get
2g=-4 => g=-2
2f=-10 => f=-5
c=4
Radius = (sqrt{g^2+f^2-c} = sqrt{4+25-4})=5.
5. Find the equation of circle which pass through (5, 9) and center at (2, 5).
a) x2+y2+4x-10y+4=0
b) x2+y2-4x-10y+4=0
c) x2+y2+4x+10y+4=0
d) x2+y2+4x-10y-4=0
Answer: b
Clarification: Equation of circle with center at (a, b) and radius r units is
(x-a)2 + (y-b)2 = r2
(5-2)2 + (9-5)2 = r2 => r2=32+42 => r=5.
So, equation of circle is (x-2)2+(y-5)2=52 => x2+y2-4x-10y+4=0.
6. If a circle pass through (2, 0) and (0, 4) and center at x-axis then find the radius of the circle.
a) 25 units
b) 20 units
c) 5 units
d) 10 units
Answer: c
Clarification: Equation of circle with center at x-axis (a, 0) and radius r units is
(x-a)2+(y)2=r2
=>(2-a)2+(0)2=r2
And (0-a)2+(4)2=r2
=>(a-2)2=a2+42 => (-2)(2a-2) =16 => a-1=-4 => a=-3
So, r2 = (2+3)2=52
r=5 units.
7. If a circle pass through (4, 0) and (0, 2) and center at y-axis then find the radius of the circle.
a) 25 units
b) 20 units
c) 5 units
d) 10 units
Answer: c
Clarification: Equation of circle with center at y-axis (0, b) and radius r units is
(x)2+(y-b)2=r2
=>(4)2+(-b)2=r2
And (0)2+(2-b)2=r2
=>(b-2)2=b2+42 => (-2)(2b-2)=16 => b-1=-4 => b=-3
So, r2=42+32=52 => r=5 units.
8. The point (1, 4) lie ___________ the circle x2+y2-2x-4y+2=0.
a) inside circle
b) outside circle
c) on the circle
d) either inside or outside
Answer: b
Clarification: Circle has equation x2+y2-2x-4y+2=0.
12+42-2*1-4*4+2 = 1+16-2-16+2 =1 > 0 so, point is outside the circle.
9. The point (0, 0) lie ___________ the circle x2+y2-2x-4y=0.
a) inside circle
b) outside circle
c) on the circle
d) either inside or outside
Answer: c
Clarification: Circle has equation x2+y2-2x-4y=0.
02+02-2*0-4*0+0 = 0 so, point is on the circle.
10. The point (6, 2) lie ___________ the circle x2+y2-2x-4y-36=0.
a) inside circle
b) outside circle
c) on the circle
d) either inside or outside
Answer: c
Clarification: Circle has equation x2+y2-2x-4y-16=0.
62+22-2*6-4*2-36 = 36+4-12-8-36 =-16<0 so, point is inside the circle.