Calculate the ratio in which the line joining `A(5, 6) and B(-3, 4)` is divided by `x = 2`. Also, find the point of intersection.

Answer

To solve the problem of finding the ratio in which the line joining points A(5, 6) and B(-3, 4) is divided by the line x = 2, and to find the point of intersection, we will follow these steps:

### Step 1: Identify the coordinates of points A and B
- Point A is given as A(5, 6).
- Point B is given as B(-3, 4).

### Step 2: Use the section formula
The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of point P can be calculated as:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

### Step 3: Set up the equation for x-coordinate
Since the line x = 2 divides the segment, we set the x-coordinate of point P to 2:
\[ 2 = \frac{m(-3) + n(5)}{m+n} \]

### Step 4: Rearrange the equation
Rearranging the equation gives:
\[ 2(m+n) = m(-3) + n(5) \]
\[ 2m + 2n = -3m + 5n \]
\[ 2m + 3m = 5n - 2n \]
\[ 5m = 3n \]

### Step 5: Express the ratio
From the equation \(5m = 3n\), we can express the ratio \(m:n\):
\[ \frac{m}{n} = \frac{3}{5} \]
Thus, the ratio in which the line segment AB is divided by the line x = 2 is 3:5.

### Step 6: Find the value of m and n
Let \(m = 3k\) and \(n = 5k\) for some constant \(k\). Therefore, the total ratio is \(m+n = 8k\).

### Step 7: Substitute m and n into the y-coordinate formula
Now, we can find the y-coordinate of point P using the section formula:
\[ y = \frac{my_2 + ny_1}{m+n} \]
Substituting \(m = 3k\), \(n = 5k\), \(y_1 = 6\), and \(y_2 = 4\):
\[ y = \frac{3k(4) + 5k(6)}{8k} \]
\[ y = \frac{12k + 30k}{8k} = \frac{42k}{8k} = \frac{42}{8} = \frac{21}{4} \]

### Step 8: Write the coordinates of point P
Thus, the coordinates of point P, where the line x = 2 intersects the line segment AB, are:
\[ P\left(2, \frac{21}{4}\right) \]

### Final Answer
- The ratio in which the line joining A and B is divided by x = 2 is **3:5**.
- The point of intersection is **P(2, 21/4)**.