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If the equations of the two diameters of a circle are x - y = 3 and 3x + y = 5 and the radius of the circle is 5, find the equation of the circle.
1. 9x2 + 4y2 = 0
2. x2 + y2 - 6x + 2y - 25 = 0
3. x2 + y2 - 3x + y + 18 = 0
4. x2 + y2 - 4x + 2y - 20 = 0

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Correct Answer - Option 4 : x2 + y2 - 4x + 2y - 20 = 0

Concept:

The equation of the circle with centre at (h, k) and radius equal to a, is

(x - h)2 + (y - k)2 = a2

Calculation:

We have, the equations of the diameters as

x - y = 3      ----(1)

3x + y = 5      ----(2)

On solving equations (1) & (2), we get

x = 2 and y = -1

Since the point of intersection of any two diameters of a circle is its centre. 

∴ Coordinates of the centre are (2, -1) and its radius is 5 (given).

So, Equation of the circle is given by

(x - 2)2 + (y + 1)2 = 52 

⇒ x2 + 4 - 4x + y2 + 1 + 2y = 25

⇒ x2 + y2 - 4x + 2y - 20 = 0

Hence, the equation of the circle is x2 + y2 - 4x + 2y - 20 = 0.

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