Circle Test

Result

Circle Test - 3

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Weekly Quiz Competition

Question 1
1 / -0

Find the number of points (x,y), having integral coordinates satisfying the condition x² + y² < 25.

A
69

B
78

C
62

D
85

Solution

Since x² + y² < 25 and x and y are integers, the possible
values of x and y ⊂ ( 0, ± 1, ± 2, ± 3, ± 4 ).

Thus x and y can be chosen in 9 ways each and ( x, y ) can be chosen in
9 × 9 = 81 ways.
However we have to exclude cases
( ± 3, ± 4 ),( ± 4, ± 3 ) and ( ± 4, ± 4 ) i.e., 3 × 4 = 12 cases.

The correct answer is: 69
Question 2
1 / -0

If one of two circle x² + y² + λ₁ ( x - y ) + c = 0, and
x² + y² + λ₂ ( x - y ) + c = 0 , where λ₁ λ₂ ∈ R , λ₁ ≠ λ₂ lies within the other then

A
c > 0

B
c < 0

C
c = 0

D
c ≥ 0

Solution

x² + y² + λ₁ (x - y) + c = 0 .....(1)
x² + y² + λ₂ (x - y) + c = 0
Radical axes (λ₁ - λ₂)(x - y) = 0
⇒ x = y
Putting in (1) 2x² + c = 0
⇒ c > 0 for non real x.

The correct answer is: c > 0
Question 3
1 / -0

The line 4x + 3y - 4 = 0, divides the circumference of the circle centered at ( 5 ,3 ) in the ratio 1 : 2 .Then the equation if the circle is

A
x² + y² – 10x – 6y – 66 = 0

B
x² + y² – 10x – 6y – 100 = 0

C
x² + y² – 10x – 6y + 100 = 0

D
x² + y² – 10x – 6y + 66 = 0

Solution

Since 4x + 3y - 4 = 0 is dividing the circumference in the ratio 1:2 angle subtended at the centre =2π/3.
Also the perpendicular distance from the centre of the given line is 5.
⇒ Radius =10
⇒ Equation of the circle is
x² + y² - 10x - 6y - 66 = 0.

The correct answer is: x² + y² – 10x – 6y – 66 = 0