Mathematics Test 82

Result

Mathematics Test 82

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Weekly Quiz Competition

Question 1
4 / -1

A variable circle which always touch the line x+ y–2=0 at (1, 1) cut the circle x² + y² + 4x + 5y – 6 = 0. All the common chord of these two circles always passes through a fix point then fix point is:-

A
(1, 1)

B
(–2, 4)

C
(6, –4)

D
None of these

Solution

Any O which touch line is

(x – 1)² + (y – 1)² + λ(x + y – 2) = 0

Common chord of both O is

(λ – 6) x + (λ – 7)y + 8 – 2λ = 0

⇒ (–6x – 7y + 8) + λ(x + y – 2) = 0

Which is family of line passes through (6, –4)
Question 2
4 / -1

A circle C₁ of radius 2 touches both x-axis and y-axis. Another circle C₂ whose radius is greater than 2 touches circle C₁ and both the axes. Then the radius of circle C₂ is-

A
6 - 4 √2

B
6 + 4 √2

C
6 - 4 √3

D
6 + 4 √3

Solution

OA² = 2r² = (OC + CA)²

2r² = OC² + CA² + 2.OC.CA

2r² = 8 + (r + 2)² + 2.2√ (r+2)

By solving this eq.

r = 6 + 4 √2
Question 3
4 / -1

If the circle x² + y² + 2gx + 2ƒy + c = 0 bisects the circumference of the circle x² + y² – 2x – 4y – 11 = 0, then the area of equilateral triangle taking common chord as side of the triangle will be :-

A
16 √3 sq. units

B
4 √3 sq. units

C
8 √3 sq. units

D
8 sq. units