Practice question on circle for iitjee , NDA and airforce | SAT | UPSC | IIT JEE | CBSE | SET 2 | 16-30

Circle

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Question no. 16

Under which of the following conditions does a general equation ax² + 2hxy + by² +2gx +2fy +c = 0 represents a circle?

looks_one h=g , a=b

looks_two h=g=f , a=b

looks_3 h=0 , a=b

looks_4 h=0 , g² + f² -c=a+b

Option looks_3 h=0 , a=b

Solution :
Condition for circle
1) a = b
2) h = 0

Question no. 17

What is the equation to circle which touches both the axes and has centre on the line x+y-4=0 ?

looks_one x² + y² +4x -4y +4 = 0

looks_two x² + y² +4x +4y +4 = 0

looks_3 x² + y² -4x -4y +4 = 0

looks_4 x² + y² -4x +4y +4 = 0

option looks_3 x² + y² -4x -4y +4 = 0

Solution :
x+ y = 4
If both the axes are touched by circle
(x-2)² + (y-2)² = 4
x² + y² -4x -4y +4 = 0

Question no. 18

If the Circle x² + y² +2gx +2fy +c = 0 (c>0) touches the y-axis , then which one of the following is correct ?

looks_one g=-√c only

looks_two g= ∓√c

looks_3 f=√c only

looks_4 f=∓√c

option looks_4 f=∓√c

Solution :
Circle is touching the y axis
(0 , y) will satisfy the equation
y² + 2fy + c = 0
Circle is touching at only one point
D = 0
4f² - 4c = 0
f² - c = 0
f = ∓√c

Question no. 19

The radius of the circle is x² + y² +2gx +2fy +c = 0 be r, then it will touch both the axes , if

looks_one g = f = r

looks_two g = f = c = r

looks_3 g=f = √c =r

looks_4 g=f and c²= r

option looks_3 g=f = √c =r

Solution :
g = f = r = √c

Question no. 20

The equation x² + y²=0 denotes

looks_one a point

looks_two a circle

looks_3 x -axis

looks_4 y- axis

option looks_one a point

Solution :
x² + y²=0 , A circle with radius 0 is a point

Question no. 21

Equation of the circle passing through origin is x² + y² -6x +2y= 0 , what is the equation of one of its diameters?

looks_one x+3y=0

looks_two x+y=0

looks_3 x=y

looks_4 3x+y=0

option looks_one x+3y=0

Solution :
x² + y² -6x +2y= 0
(3,-1) is the center
x+3y=0 is satisfied with point ( 3 , -1)

Question no. 22

Center of circle (x-3)² +(y-4)²=5 is

looks_one (3,4)

looks_two (-3,-4)

looks_3 (4,3)

looks_4 (-4,-3)

option looks_one (3,4)

Solution :
(x-3)² +(y-4)²=5
(3,4) is the center of circle

Question no. 23

The equation of the circle passing through the points (0,0) , (0,b) and (a ,b) is

looks_one x² + y² +ax +by= 0

looks_two x² + y² -ax +by= 0

looks_3 x² + y² -ax -by= 0

looks_4 x² + y² +ax -by= 0

option looks_3 x² + y² -ax -by= 0

Solution :
x² + y² + 2fy + 2gx + c = 0
For (0 ,0) : c = 0
For (a,0) : a² + 2ga = 0
g = -a/2
For (0,b) : f = -b/2
x² + y² -ax -by= 0

Question no. 24

Which of the following line is a diameter of the circle x² +y² -6x -8y -9 = 0

looks_one 3x-4y =0

looks_two 4x-3y =9

looks_3 x+y=7

looks_4 x-y=1

option looks_3 x+y=7

Solution :
x² +y² -6x -8y -9 = 0
(3 , 4) is the circle's center
only x+y=7 is satisfied with point (3,4)

Question no. 25

The equation of circle which touches the axes at a distance 5 origin is x² +y² -2ay -2ax + a² = 0 touches

looks_one only y-axis

looks_two only x-axis

looks_3 both the axes

looks_4 neither of the axes

option looks_3 both the axes

Solution :
x² +y² -2ay -2ax + a² = 0
(x-a)² + (y-a)² = a²
it will touch both the axes

Question no. 26

For the circle x² +y² +3x +3y = 0 , which of the following statements is true

looks_one circle lies on x-axis

looks_two circle is at origin

looks_3 circle lies on y-axis

looks_4 circle passes through origin

option looks_4 circle passes through origin

Solution :
x² +y² +3x +3y = 0
(x - 3/2)² + (y-3/2)² = 9/2
(0,0) will satisfy the equation , therefore circle passes through the origin

Question no. 27

The equation of the circle which passes through the origin and cuts off intercepts of 2 units length from the negative co-ordinate axes, is

looks_one x² +y² - 2x + 2y =0

looks_two x² +y² + 2x + 2y =0

looks_3 x² +y² - 2x - 2y =0

looks_4 x² +y² + 2x - 2y =0

option looks_two x² +y² + 2x + 2y =0

Solution :

Center will lie in 3rd quadrant
x² +y² + 2x + 2y =0

Question no. 28

Point (1,2) relative to the circle x² +y² + 4x - 2y -4 =0 is a/an

looks_one exterior point

looks_two interior point , but not center

looks_3 boundary point

looks_4 centre

option looks_one exterior point

Solution :
x² +y² + 4x - 2y -4 =0
put the point (1,2) in equation of circle
1² +2² + 4(1) - 2(2) -4 = 1 > 0
Therefore , it is exterior point

Question no. 29

What is the radius of the circle touching x-axis at (3,0) and y-axis at (0,3) ?

looks_one 3 units

looks_two 4 units

looks_3 5 units

looks_4 6 units

option looks_one 3 units

Solution :
If circle touchs both the axis at same distance
Therefore , g = f = r
r = 3 unit

Question no. 30

Which one of the following point lies inside a circle of radius 6 and centre at (3,5)?

looks_one (-2,-1)

looks_two (0,1)

looks_3 (-1,-2)

looks_4 (2 ,-1)

option looks_two (0,1)

Solution :
(x-3)² + (y-5)² = 6²
(x-3)² + (y-5)² - 36
(0-3)² + (1-5)² - 36 = 9 + 16 - 36 = -9 < 0

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