A line intersects the straight lines 5x - y - 4 = 0 and 3x - 4y - 4 = 0 at A and B respectively. If a point P (1, 5) on the line AB is such that AP : PB = 2:1 (internally), find the point A.

Options

A. `((75)/(17).(307)/(17))`
B. `((65)/(17),(304)/(17))`
C. `((75)/(17),(104)/(17))`
D. `((75)/(17),(180)/(17))`

Answer

To find the point A where the line intersects the line equations \(5x - y - 4 = 0\) and \(3x - 4y - 4 = 0\), we can follow these steps:

### Step 1: Identify the equations of the lines
The two lines are given as:
1. \(5x - y - 4 = 0\) (Line 1)
2. \(3x - 4y - 4 = 0\) (Line 2)

### Step 2: Express y in terms of x for both lines
For Line 1:
\[ y = 5x - 4 \]

For Line 2:
\[ 4y = 3x - 4 \implies y = \frac{3}{4}x - 1 \]

### Step 3: Let the coordinates of point A be \((t, 5t - 4)\)
Assuming point A lies on Line 1, we can express point A as:
\[ A(t, 5t - 4) \]

### Step 4: Let the coordinates of point B be \((x_2, y_2)\)
Assuming point B lies on Line 2, we can express point B as:
\[ B(x_2, y_2) \]

### Step 5: Use the section formula to find coordinates of P
Given that point P (1, 5) divides the segment AB in the ratio \(AP:PB = 2:1\), we can use the section formula:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
where \(m = 2\), \(n = 1\), \(x_1 = t\), \(y_1 = 5t - 4\), \(x_2\) and \(y_2\) are the coordinates of point B.

### Step 6: Set up equations using the coordinates of P
From the x-coordinates:
\[ 1 = \frac{2x_2 + t}{3} \implies 3 = 2x_2 + t \implies 2x_2 = 3 - t \implies x_2 = \frac{3 - t}{2} \]

From the y-coordinates:
\[ 5 = \frac{2y_2 + (5t - 4)}{3} \implies 15 = 2y_2 + 5t - 4 \implies 2y_2 = 19 - 5t \implies y_2 = \frac{19 - 5t}{2} \]

### Step 7: Substitute \(x_2\) and \(y_2\) into Line 2 equation
Substituting \(x_2\) and \(y_2\) into the equation of Line 2:
\[ 3\left(\frac{3 - t}{2}\right) - 4\left(\frac{19 - 5t}{2}\right) - 4 = 0 \]

### Step 8: Simplify the equation
Multiply through by 2 to eliminate the fraction:
\[ 3(3 - t) - 4(19 - 5t) - 8 = 0 \]
\[ 9 - 3t - 76 + 20t - 8 = 0 \]
\[ 17t - 75 = 0 \implies t = \frac{75}{17} \]

### Step 9: Find the coordinates of point A
Now substituting \(t\) back into the expression for point A:
\[ A\left(\frac{75}{17}, 5\left(\frac{75}{17}\right) - 4\right) = \left(\frac{75}{17}, \frac{375}{17} - \frac{68}{17}\right) = \left(\frac{75}{17}, \frac{307}{17}\right) \]

### Final Answer
The coordinates of point A are:
\[ A\left(\frac{75}{17}, \frac{307}{17}\right) \]