Let `C` be a curve which is the locus of the point of intersection of lines `x=2+m` and `m y=4-mdot` A circle `s-=(x-2)^2+(y+1)^2=25` intersects the curve `C` at four points: `P ,Q ,R ,a n dS` . If `O` is center of the curve `C ,` then `O P^2+O P^2+O R^2+O S^2` is (a) `50` (b) `100` (c) `25` (d) `(25)/2`

Options

A. 50
B. 100
C. 25
D. 25/5

Explanation

To solve the problem step by step, we will first derive the equation of the curve \( C \) and then find the required sum of squares of distances from the center \( O \) to the points of intersection \( P, Q, R, S \).

### Step 1: Find the equations of the lines
The lines given are:
1. \( x = 2 + m \)
2. \( my = 4 - m \)

From the first equation, we can express \( m \) in terms of \( x \):
\[ m = x - 2 \]

Substituting this value of \( m \) into the second equation:
\[ (x - 2)y = 4 - (x - 2) \]
\[ (x - 2)y = 6 - x \]

### Step 2: Rearranging to find the equation of the curve
Rearranging gives:
\[ (x - 2)y + x - 6 = 0 \]
This can be rewritten as:
\[ xy - 2y + x - 6 = 0 \]
This is the equation of a hyperbola.

### Step 3: Find the equation of the circle
The given circle is:
\[ (x - 2)^2 + (y + 1)^2 = 25 \]

### Step 4: Find points of intersection
To find the points of intersection between the hyperbola and the circle, we can substitute \( y \) from the hyperbola equation into the circle equation. However, we can also use the fact that the circle has a radius of \( 5 \) (since \( 25 = 5^2 \)).

### Step 5: Calculate the center of the circle
The center of the circle is at the point \( (2, -1) \).

### Step 6: Calculate the distances from the center to the points of intersection
Let \( O \) be the center of the circle at point \( (2, -1) \). The distance from \( O \) to any point \( P, Q, R, S \) on the circle is equal to the radius of the circle, which is \( 5 \).

### Step 7: Sum of the squares of the distances
The sum of the squares of the distances from \( O \) to the points \( P, Q, R, S \) can be calculated as:
\[ OP^2 + OQ^2 + OR^2 + OS^2 = 4 \times (radius)^2 = 4 \times 5^2 = 4 \times 25 = 100 \]

### Final Answer
Thus, the value of \( OP^2 + OQ^2 + OR^2 + OS^2 \) is \( 100 \).