Consider the following in respect of the circle 4x^{2}+4y^{2}-4ax-4ay+a^{2}=0 :<br>1. The circle touches <strong>BOTH</strong> the axes.<br>2. The diameter of the circle is 2a.<br>3. The centre of the circle lies on the line x+y=a.<br>How many of the statements given above are correct?

Explanation

Divide the equation by 4 to get standard form: x^2 + y^2 - ax - ay + \frac{a^2}{4} = 0. The center is (\frac{a}{2}, \frac{a}{2}) and the radius is r = \sqrt{(\frac{a}{2})^2 + (\frac{a}{2})^2 - \frac{a^2}{4}} = \frac{|a|}{2}. Since the magnitude of both x and y coordinates of the center equals the radius, it touches both axes (Statement 1 is true). The diameter is 2r = |a|, not 2a (Statement 2 is false). The center satisfies x+y = \frac{a}{2} + \frac{a}{2} = a (Statement 3 is true). Therefore, only two statements are correct.

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