(i) Find the equation of a circle concentric with the circle `x^(2)+y^(2)-8x+6y-10=0` and passes through the point (-2,3).
(ii) Find the equation of circle concentric with the circle `x^(2)+y^(2)-4x-8y-6=0` and whose radius is three times the radius of this circle.

Answer

To solve the given problems step by step, we will break down each part of the question.

### Part (i)

**Step 1: Identify the given circle's equation.**
The equation of the circle is:
\[ x^2 + y^2 - 8x + 6y - 10 = 0 \]

**Step 2: Rewrite the equation in standard form.**
To find the center and radius, we can rewrite the equation in standard form:
\[ (x^2 - 8x) + (y^2 + 6y) = 10 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 - 8x + 16) + (y^2 + 6y + 9) = 10 + 16 + 9 \]
This simplifies to:
\[ (x - 4)^2 + (y + 3)^2 = 35 \]
Thus, the center is \((4, -3)\) and the radius is \(\sqrt{35}\).

**Step 3: Write the equation of the concentric circle.**
A circle concentric with this one will have the same center but a different radius. The general form of the equation for a concentric circle is:
\[ (x - 4)^2 + (y + 3)^2 = r^2 \]
where \(r\) is the radius of the new circle.

**Step 4: Substitute the point (-2, 3) into the equation.**
Since the new circle passes through the point (-2, 3):
\[ (-2 - 4)^2 + (3 + 3)^2 = r^2 \]
Calculating this gives:
\[ (-6)^2 + (6)^2 = r^2 \]
\[ 36 + 36 = r^2 \]
\[ r^2 = 72 \]

**Step 5: Write the final equation of the circle.**
Thus, the equation of the required circle is:
\[ (x - 4)^2 + (y + 3)^2 = 72 \]

### Part (ii)

**Step 1: Identify the given circle's equation.**
The equation of the circle is:
\[ x^2 + y^2 - 4x - 8y - 6 = 0 \]

**Step 2: Rewrite the equation in standard form.**
Rearranging gives:
\[ (x^2 - 4x) + (y^2 - 8y) = 6 \]
Completing the square:
\[ (x^2 - 4x + 4) + (y^2 - 8y + 16) = 6 + 4 + 16 \]
This simplifies to:
\[ (x - 2)^2 + (y - 4)^2 = 26 \]
Thus, the center is \((2, 4)\) and the radius is \(\sqrt{26}\).

**Step 3: Find the radius of the new circle.**
The radius of the new circle is three times the radius of the original circle:
\[ \text{New radius} = 3 \times \sqrt{26} = 3\sqrt{26} \]

**Step 4: Write the equation of the new concentric circle.**
The equation for the new circle is:
\[ (x - 2)^2 + (y - 4)^2 = (3\sqrt{26})^2 \]
Calculating this gives:
\[ (x - 2)^2 + (y - 4)^2 = 9 \times 26 = 234 \]

**Final Equation:**
Thus, the equation of the required circle is:
\[ (x - 2)^2 + (y - 4)^2 = 234 \]