Answer: a = 10
radius = 7
Explanation:
The generic equation for a circle of radius R is, centered in the point (a, b)
(x - a)^2 + (x - b)^2 = R^2
In this case, we know that the center is (3, -5)
Then the equation of the circle must be something like:
(x - 3)^2 + (y + 5)^2 = R^2
And we have:
x^2 + y^2 - 6*x + a*y = 15
First, let's complete squares for x:
(x - 3)^2 = x^2 + 2*(-3)*x + (-3)^2 = x^2 - 6*x + 9.
Then we can add 9 to both sides of the equation to get:
x^2 + y^2 - 6*x + a*y + 9 = 15 + 9
(x^2 - 6*x + 9) + y^2 + a*y = 24
(x - 3)^2 + y^2 + a*y = 24
Now let's do the same for y:
(y + 5)^2 = y^2 + 2*5*y + 5^2 = y^2 + 10*y + 25
From this we can already see that a must be equal to 10, if we replace a by 10, the circle equation becomes:
(x - 3)^2 + y^2 + 10*y = 24
Now we can add 25 to both sides of the equation to get:
(x - 3)^2 + y^2 + 10*y + 25 = 24 + 25
(x - 3)^2 + (y + 5)^2 = 49
An the radius of the circle will be equal to:
r = √49 = 7