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The photo contains a mathematical problem and its solution related to geometry. Here's a breakdown of the content:

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### Problem:
Given two points \( A(2,0) \) and \( B(4, a) \), if \( P(x, y) \) lies on a hyperbola with a horizontal transverse axis, and the ratio of the distances \( \frac{PA}{PB} = \frac{1}{2} \), find the equation of the hyperbola that passes through the point \( C \).

### Approach:
1. **Step 1: Distance Ratio Equation**  
   Use the distance formula to express the distances \( PA \) and \( PB \) in terms of the coordinates of points \( A \), \( B \), and \( P \).  
   The distance formula is:
   \[
   PA = \sqrt{(x - 2)^2 + y^2}
   \]
   \[
   PB = \sqrt{(x - 4)^2 + (y - a)^2}
   \]
   Given that \( \frac{PA}{PB} = \frac{1}{2} \), this ratio can be set up as an equation and squared to remove the square roots.

2. **Step 2: Algebraic Manipulation**  
   After squaring and simplifying the resulting equation, expressions involving the coordinates \( x \), \( y \), and \( a \) emerge, leading to the equation of the hyperbola.

3. **Step 3: Substitution and Hyperbola Equation**  
   The derived equation involves identifying constants and substituting the required conditions to find the hyperbola equation.

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Would you like me to provide more detailed steps for solving this problem, including how to work with the distance formula and ratios to derive the final equation? Also, here are five related questions:

1. How is the distance formula derived for two points in a plane?
2. What is the general form of a hyperbola with a horizontal transverse axis?
3. How do you solve equations involving square roots and ratios?
4. What is the significance of the condition \( \frac{PA}{PB} = \frac{1}{2} \) in determining the curve?
5. How can this problem be extended to work with other conic sections like ellipses?

**Tip:** When dealing with distance ratio problems, squaring the equation is a common technique to eliminate square roots, making the algebraic manipulation easier.

Diketahui titik A(2,0) dan B(4,a). Jika P(x,y) terletak pada garis lengkung berbentuk hiperbola sumbu mendatar, dengan perbandingan jarak PA/PB = 1/2. Tentukan persamaan hiperbola yang memotong C.

Learn how to derive the equation of a hyperbola given two points A(2,0) and B(4,a), with a distance ratio of PA/PB = 1/2. This problem covers key concepts in geometry, including the distance formula and the standard form of a hyperbola equation. Suitable for students in Grades 10-12.

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