Find the value of K if the points
(i) (4,K) and (2,3) are conjugate with respect to the circle `x^(2)+y^(2)=17`.
(ii) (4,2) and (K,-3) are conjugate with respect to the circle `x^(2)+y^(2)-5x+8y+6=0`

Answer

To find the value of K for the given points that are conjugate with respect to the specified circles, we will solve each part step by step.

### Part (i)

**Given:**
- Points: (4, K) and (2, 3)
- Circle: \( x^2 + y^2 = 17 \)

**Step 1: Write the condition for conjugate points.**
For two points (x1, y1) and (x2, y2) to be conjugate with respect to a circle, the condition is given by:
\[ S_1 = 0 \]
where \( S_1 = x_1 x + y_1 y - r^2 \) (r^2 is the radius squared).

**Step 2: Substitute the values into the equation.**
Here, \( x_1 = 4, y_1 = K, x_2 = 2, y_2 = 3 \) and \( r^2 = 17 \).
Thus, the equation becomes:
\[ 4 \cdot 2 + K \cdot 3 - 17 = 0 \]

**Step 3: Simplify the equation.**
Calculating gives:
\[ 8 + 3K - 17 = 0 \]
\[ 3K - 9 = 0 \]

**Step 4: Solve for K.**
\[ 3K = 9 \]
\[ K = 3 \]

### Part (ii)

**Given:**
- Points: (4, 2) and (K, -3)
- Circle: \( x^2 + y^2 - 5x + 8y + 6 = 0 \)

**Step 1: Rewrite the circle equation in standard form.**
We can complete the square for the circle equation:
\[ (x^2 - 5x) + (y^2 + 8y) + 6 = 0 \]
Completing the square:
\[ (x - \frac{5}{2})^2 - \frac{25}{4} + (y + 4)^2 - 16 + 6 = 0 \]
This simplifies to:
\[ (x - \frac{5}{2})^2 + (y + 4)^2 = \frac{25}{4} + 10 = \frac{65}{4} \]

**Step 2: Write the polar equation for the point (4, 2).**
The polar of point (x1, y1) with respect to the circle is given by:
\[ S_1 = 0 \]
Where:
\[ S_1 = 4x + 2y - (5 \cdot 4 + 8 \cdot 2 + 6) \]

Calculating gives:
\[ 4x + 2y - (20 + 16 + 6) = 0 \]
\[ 4x + 2y - 42 = 0 \]

**Step 3: Substitute (K, -3) into the polar equation.**
Substituting \( x = K \) and \( y = -3 \):
\[ 4K + 2(-3) - 42 = 0 \]
\[ 4K - 6 - 42 = 0 \]
\[ 4K - 48 = 0 \]

**Step 4: Solve for K.**
\[ 4K = 48 \]
\[ K = 12 \]

### Final Answers
- (i) \( K = 3 \)
- (ii) \( K = 12 \)