Find the ratio in which the line joining the points (6, 12) and (4, 9) is divided by the curve `x^(2)+y^(2)=4`.

Answer

To solve the problem of finding the ratio in which the line joining the points (6, 12) and (4, 9) is divided by the curve \(x^2 + y^2 = 4\), we can follow these steps:

### Step 1: Define the Points and the Ratio
Let the points be:
- \(A(6, 12)\)
- \(B(4, 9)\)

Let the point \(P\) divide the line segment \(AB\) in the ratio \(k:1\). Therefore, the coordinates of point \(P\) can be expressed using the section formula.

### Step 2: Use the Section Formula
The coordinates of point \(P\) can be calculated as:
\[ P\left(\frac{4k + 6}{k + 1}, \frac{9k + 12}{k + 1}\right) \]

### Step 3: Substitute into the Curve Equation
Since point \(P\) lies on the curve \(x^2 + y^2 = 4\), we substitute the coordinates of \(P\) into this equation:
\[ \left(\frac{4k + 6}{k + 1}\right)^2 + \left(\frac{9k + 12}{k + 1}\right)^2 = 4 \]

### Step 4: Expand the Equation
Expanding both terms:
1. For the first term:
\[ \left(\frac{4k + 6}{k + 1}\right)^2 = \frac{(4k + 6)^2}{(k + 1)^2} = \frac{16k^2 + 48k + 36}{(k + 1)^2} \]
2. For the second term:
\[ \left(\frac{9k + 12}{k + 1}\right)^2 = \frac{(9k + 12)^2}{(k + 1)^2} = \frac{81k^2 + 216k + 144}{(k + 1)^2} \]

Combining these:
\[ \frac{16k^2 + 48k + 36 + 81k^2 + 216k + 144}{(k + 1)^2} = 4 \]

### Step 5: Simplify the Equation
Combine the terms in the numerator:
\[ \frac{97k^2 + 264k + 180}{(k + 1)^2} = 4 \]

Cross-multiplying gives:
\[ 97k^2 + 264k + 180 = 4(k + 1)^2 \]
Expanding the right side:
\[ 4(k^2 + 2k + 1) = 4k^2 + 8k + 4 \]

### Step 6: Rearrange the Equation
Setting the equation to zero:
\[ 97k^2 + 264k + 180 - 4k^2 - 8k - 4 = 0 \]
This simplifies to:
\[ 93k^2 + 256k + 176 = 0 \]

### Step 7: Factor the Quadratic Equation
We can factor this quadratic equation. Using middle term splitting:
\[ (3k + 4)(31k + 44) = 0 \]
Setting each factor to zero gives:
1. \(3k + 4 = 0 \Rightarrow k = -\frac{4}{3}\)
2. \(31k + 44 = 0 \Rightarrow k = -\frac{44}{31}\)

### Step 8: Determine the Ratios
The ratios in which the line is divided are:
- \( \frac{4}{3} \) (external division)
- \( \frac{44}{31} \) (external division)

### Final Answer
Thus, the required ratios are \( \frac{4}{3} \) and \( \frac{44}{31} \).

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