If each of the points `(x_1,4),(-2,y_1)` lies on the line joining the points `(2,-1)a n d(5,-3)` , then the point `P(x_1,y_1)` lies on the line. (a) `6(x+y)-25=0` (b) `2x+6y+1=0` (c)`2x+3y-6=0` (d) `6(x+y)+25=0`

Options

A. `6(x+y)-25 = 0`
B. `2x+6y+1 =0`
C. `2x+3y-6=0`
D. `6(x+y)+25=0`

Explanation

To solve the problem, we need to find the equation of the line that passes through the points \( (2, -1) \) and \( (5, -3) \) and then check if the points \( (x_1, 4) \) and \( (-2, y_1) \) lie on this line. Finally, we will see if the point \( P(x_1, y_1) \) lies on any of the given options.

### Step 1: Find the slope of the line
The slope \( m \) of the line passing through the points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the points \( (2, -1) \) and \( (5, -3) \):
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