If the circles of same radius and centres at (2,3) and (5,6) cut orthogonally, then find a.

Steps to Solve

1. Identify Circle Equations

The centers of the circles are given as ( (2, 3) ) and ( (5, 6) ). Let the common radius be ( r ). The equations of the circles can be written as:

$$ (x - 2)^2 + (y - 3)^2 = r^2 $$

$$ (x - 5)^2 + (y - 6)^2 = r^2 $$

2. Set Up the Condition for Orthogonality

For two circles to intersect orthogonally, the following condition must be satisfied:

$$ 2r^2 = d^2 $$

where ( d ) is the distance between the centers of the two circles.

3. Calculate the Distance Between the Centers

The distance ( d ) between the centers ( (2, 3) ) and ( (5, 6) ) can be calculated using the distance formula:

$$ d = \sqrt{(5 - 2)^2 + (6 - 3)^2} $$

Calculating this yields:

$$ d = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} $$

4. Find ( d^2 )

Now, compute ( d^2 ):

$$ d^2 = (3\sqrt{2})^2 = 18 $$

5. Use the Orthogonality Condition

Substituting ( d^2 ) into the orthogonality condition gives us:

$$ 2r^2 = 18 $$

Solving for ( r^2 ):

$$ r^2 = \frac{18}{2} = 9 $$

Thus, ( r = 3 ).

6. Find the Value of ( a )

The radius ( r ) is also mentioned as ( a ), therefore:

$$ a = 3 $$