Question

If one of the diameters of the circle, given by the equation ( x^2 + y^2 - 4x + 6y - 12 = 0 ), is a chord of a circle, 'S', whose centre is at ( (-3, 2) ), then the length of radius of 'S' is ________ units.

• A. 5
• B. ( 5\sqrt{2} )
• C. ( 5\sqrt{3} )
• D. 10

Hint:

For a circle where a diameter is a chord of another circle, the distance between centres, radius of the first circle, and radius of the second circle form a right-angled triangle.

Answer 1

The Correct Option is C

Step 1: Find centre and radius of the first circle
Equation: ( x^2 + y^2 - 4x + 6y - 12 = 0 ).
Centre (C_1 = (2, -3)) and radius (r_1 = \sqrt{2^2 + (-3)^2 - (-12)} = \sqrt{4 + 9 + 12} = 5).

Step 2: Geometry of the chord
The diameter of the first circle (length (2r_1 = 10)) is a chord of circle 'S'.
The midpoint of this chord is the centre of the first circle, (C_1(2, -3)).

Step 3: Calculate distance between centres
Centre of 'S' is (C_s(-3, 2)).
Distance (d = \sqrt{(2 - (-3))^2 + (-3 - 2)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{50} = 5\sqrt{2}).

Step 4: Use Pythagoras theorem for radius of 'S'
(R^2 = d^2 + r_1^2 = (5\sqrt{2})^2 + 5^2 = 50 + 25 = 75).
(R = \sqrt{75} = 5\sqrt{3}).
Final Answer: (C)