This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Conic Sections – Circle”.

1. Find the equation of circle with center at origin and radius 5 units.

a) x^{2}+y^{2}=25

b) x^{2}+y^{2}=5

c) x^{2}=25

d) y^{2}=25

View Answer

Explanation: Equation of circle with center at (a, b) and radius r units is

(x-a)

^{2}+(y-b)

^{2}=r

^{2}

So, equation of circle is (x-0)

^{2}+(y-0)

^{2}=5

^{2}=> x

^{2}+y

^{2}=25.

2. Find the equation of circle with center at (2, 5) and radius 5 units.

a) x^{2}+y^{2}+4x-10y+4=0

b) x^{2}+y^{2}-4x-10y+4=0

c) x^{2}+y^{2}+4x+10y+4=0

d) x^{2}+y^{2}+4x-10y-4=0

View Answer

Explanation: Equation of circle with center at (a, b) and radius r units is

(x-a)

^{2}+(y-b)

^{2}=r

^{2}

So, equation of circle is (x-2)

^{2}+(y-5)

^{2}=5

^{2}=> x

^{2}+y

^{2}-4x-10y+4=0.

3. Find the center of the circle with equation x^{2}+y^{2}-4x-10y+4=0.

a) (-2, 5)

b) (-2, -5)

c) (2, -5)

d) (2, 5)

View Answer

Explanation: Comparing the equation with general form x

^{2}+y

^{2}+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 x^{2}+y^{2}-4x-10y+4=0.

a) 25 units

b) 20 units

c) 5 units

d) 10 units

View Answer

Explanation: Comparing the equation with general form x

^{2}+y

^{2}+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) x^{2}+y^{2}+4x-10y+4=0

b) x^{2}+y^{2}-4x-10y+4=0

c) x^{2}+y^{2}+4x+10y+4=0

d) x^{2}+y^{2}+4x-10y-4=0

View Answer

Explanation: Equation of circle with center at (a, b) and radius r units is

(x-a)

^{2}+ (y-b)

^{2}= r

^{2}

(5-2)

^{2}+ (9-5)

^{2}= r

^{2}=> r

^{2}=3

^{2}+4

^{2}=> r=5.

So, equation of circle is (x-2)

^{2}+(y-5)

^{2}=5

^{2}=> x

^{2}+y

^{2}-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

View Answer

Explanation: Equation of circle with center at x-axis (a, 0) and radius r units is

(x-a)

^{2}+(y)

^{2}=r

^{2}

=>(2-a)

^{2}+(0)

^{2}=r

^{2}

And (0-a)

^{2}+(4)

^{2}=r

^{2}

=>(a-2)

^{2}=a

^{2}+4

^{2}=> (-2)(2a-2) =16 => a-1=-4 => a=-3

So, r

^{2}= (2+3)

^{2}=5

^{2}

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

View Answer

Explanation: Equation of circle with center at y-axis (0, b) and radius r units is

(x)

^{2}+(y-b)

^{2}=r

^{2}

=>(4)

^{2}+(-b)

^{2}=r

^{2}

And (0)

^{2}+(2-b)

^{2}=r

^{2}

=>(b-2)

^{2}=b

^{2}+4

^{2}=> (-2)(2b-2)=16 => b-1=-4 => b=-3

So, r

^{2}=4

^{2}+3

^{2}=5

^{2}=> r=5 units.

8. The point (1, 4) lie ___________ the circle x^{2}+y^{2}-2x-4y+2=0.

a) inside circle

b) outside circle

c) on the circle

d) either inside or outside

View Answer

Explanation: Circle has equation x

^{2}+y

^{2}-2x-4y+2=0.

1

^{2}+4

^{2}-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 x^{2}+y^{2}-2x-4y=0.

a) inside circle

b) outside circle

c) on the circle

d) either inside or outside

View Answer

Explanation: Circle has equation x

^{2}+y

^{2}-2x-4y=0.

0

^{2}+0

^{2}-2*0-4*0+0 = 0 so, point is on the circle.

10. The point (6, 2) lie ___________ the circle x^{2}+y^{2}-2x-4y-36=0.

a) inside circle

b) outside circle

c) on the circle

d) either inside or outside

View Answer

Explanation: Circle has equation x

^{2}+y

^{2}-2x-4y-16=0.

6

^{2}+2

^{2}-2*6-4*2-36 = 36+4-12-8-36 =-16<0 so, point is inside the circle.

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