Step 1: Understanding the Question:
A general second-degree equation \( f(x,y)=0 \) represents a pair of lines. The point of intersection is obtained by solving the equations formed by partial derivatives with respect to \( x \) and \( y \).
Step 2: Key Formula or Approach:
1. \( \frac{\partial f}{\partial x} = 0 \)
2. \( \frac{\partial f}{\partial y} = 0 \)
Step 3: Detailed Explanation:
Given: \( f(x,y) = x^2 - Ky^2 - 4x + 6y - 5 = 0 \).
Partially differentiate with respect to \( x \):
\[ \frac{\partial f}{\partial x} = 2x - 4 = 0 \Rightarrow x = 2 \]
Now, determine the value of \( K \) using the condition for a pair of lines (\( \Delta = 0 \)):
For \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \):
\( a=1, b=-K, c=-5, g=-2, f=3, h=0 \).
\( \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \).
\[ (1)(-K)(-5) + 0 - 1(3)^2 - (-K)(-2)^2 - 0 = 0 \]
\[ 5K - 9 + 4K = 0 \Rightarrow 9K = 9 \Rightarrow K = 1 \].
Now, partially differentiate with respect to \( y \) with \( K=1 \):
\[ \frac{\partial f}{\partial y} = -2(1)y + 6 = 0 \Rightarrow -2y = -6 \Rightarrow y = 3 \].
The intersection point is \( (2, 3) \).
Step 4: Final Answer:
The point of intersection is \( (2, 3) \).