(i) Find the length of the chord intercepted by the circle `x^(2)+y^(2)-8x-2y-8=0` on the line `x+y+1=0`
(ii) Find the length of the chord intercepted by the circle `x^(2)+y^(2)+8x-4y-16=0` on the line `3x-y+4=0`
(iii) Find the length of the chord formed by `x^(2)+y^(2)=a^(2)` on the line `xcos alpha +y sin alpha=p`

Answer

To solve the given problems step by step, let's break down each part of the question.

### Part (i)
**Find the length of the chord intercepted by the circle \(x^2 + y^2 - 8x - 2y - 8 = 0\) on the line \(x + y + 1 = 0\).**

**Step 1: Convert the circle's equation to standard form.**
The given equation is:
\[ x^2 + y^2 - 8x - 2y - 8 = 0 \]
We can rearrange it by completing the square.

For \(x\):
\[ x^2 - 8x \rightarrow (x - 4)^2 - 16 \]

For \(y\):
\[ y^2 - 2y \rightarrow (y - 1)^2 - 1 \]

Substituting back, we have:
\[ (x - 4)^2 - 16 + (y - 1)^2 - 1 - 8 = 0 \]
\[ (x - 4)^2 + (y - 1)^2 - 25 = 0 \]
Thus, the standard form of the circle is:
\[ (x - 4)^2 + (y - 1)^2 = 25 \]
This indicates a circle with center \((4, 1)\) and radius \(5\).

**Step 2: Find the intersection points with the line.**
The line equation is:
\[ x + y + 1 = 0 \implies y = -x - 1 \]

Substituting \(y\) in the circle's equation:
\[ (x - 4)^2 + (-x - 1 - 1)^2 = 25 \]
\[ (x - 4)^2 + (-x - 2)^2 = 25 \]
Expanding this:
\[ (x - 4)^2 + (x + 2)^2 = 25 \]
\[ (x^2 - 8x + 16) + (x^2 + 4x + 4) = 25 \]
Combining like terms:
\[ 2x^2 - 4x + 20 = 25 \]
\[ 2x^2 - 4x - 5 = 0 \]
Dividing by 2:
\[ x^2 - 2x - \frac{5}{2} = 0 \]

**Step 3: Solve the quadratic equation.**
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-\frac{5}{2})}}{2(1)} \]
\[ = \frac{2 \pm \sqrt{4 + 10}}{2} = \frac{2 \pm \sqrt{14}}{2} \]
\[ = 1 \pm \frac{\sqrt{14}}{2} \]

**Step 4: Find corresponding \(y\) values.**
Using \(y = -x - 1\):
For \(x_1 = 1 + \frac{\sqrt{14}}{2}\):
\[ y_1 = -\left(1 + \frac{\sqrt{14}}{2}\right) - 1 = -2 - \frac{\sqrt{14}}{2} \]
For \(x_2 = 1 - \frac{\sqrt{14}}{2}\):
\[ y_2 = -\left(1 - \frac{\sqrt{14}}{2}\right) - 1 = -2 + \frac{\sqrt{14}}{2} \]

**Step 5: Calculate the length of the chord.**
Using the distance formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting:
\[ = \sqrt{\left(1 - \frac{\sqrt{14}}{2} - \left(1 + \frac{\sqrt{14}}{2}\right)\right)^2 + \left(-2 + \frac{\sqrt{14}}{2} - \left(-2 - \frac{\sqrt{14}}{2}\right)\right)^2} \]
\[ = \sqrt{\left(-\sqrt{14}\right)^2 + \left(\sqrt{14}\right)^2} = \sqrt{14 + 14} = \sqrt{28} = 2\sqrt{7} \]

**Final Answer for Part (i):** The length of the chord is \(2\sqrt{7}\).