If the centre of the circle passing through the origin and the point of intersecton of the pair of straingth lines `xy - 7x + 3y -21=0 ` with the coordinate axes lies on the x + y =k, then k is equal to.

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem, we need to find the value of \( k \) such that the center of the circle passing through the origin and the intersection of the given pair of straight lines lies on the line \( x + y = k \).

### Step 1: Find the intersection points of the lines with the coordinate axes

The given equation of the pair of straight lines is:
\[ xy - 7x + 3y - 21 = 0 \]

To find the intersection points with the coordinate axes, we can rewrite the equation in a more manageable form. We can factor it as follows:
\[ xy - 7x + 3y - 21 = 0 \implies (x + 3)(y - 7) = 0 \]
This gives us two lines:
1. \( x + 3 = 0 \) or \( x = -3 \)
2. \( y - 7 = 0 \) or \( y = 7 \)

Now, we find the points where these lines intersect the axes:
- The line \( x = -3 \) intersects the y-axis at \( (0, 7) \).
- The line \( y = 7 \) intersects the x-axis at \( (-3, 0) \).

### Step 2: Identify the points of intersection

The points of intersection with the axes are:
- \( A(0, 7) \)
- \( B(-3, 0) \)
- The origin \( O(0, 0) \)

### Step 3: Determine the center of the circle

Let the center of the circle be \( C(h, k) \). The circle passes through the points \( O(0, 0) \), \( A(0, 7) \), and \( B(-3, 0) \).

### Step 4: Set up the equations based on distances

The distances from the center \( C(h, k) \) to each of the points must be equal since they all lie on the circle.

1. Distance from \( C \) to \( O \):
\[ \sqrt{h^2 + k^2} \]

2. Distance from \( C \) to \( A(0, 7) \):
\[ \sqrt{(h - 0)^2 + (k - 7)^2} = \sqrt{h^2 + (k - 7)^2} \]

3. Distance from \( C \) to \( B(-3, 0) \):
\[ \sqrt{(h + 3)^2 + k^2} \]

### Step 5: Set the distances equal to each other

Equating the distance from \( C \) to \( O \) and \( A \):
\[ h^2 + k^2 = h^2 + (k - 7)^2 \]
Expanding and simplifying:
\[ k^2 = k^2 - 14k + 49 \implies 14k = 49 \implies k = \frac{49}{14} = 3.5 \]

Now, equating the distance from \( C \) to \( O \) and \( B \):
\[ h^2 + k^2 = (h + 3)^2 + k^2 \]
Expanding and simplifying:
\[ h^2 + k^2 = h^2 + 6h + 9 + k^2 \implies 0 = 6h + 9 \implies h = -\frac{3}{2} \]

### Step 6: Find \( k \)

Now we have:
- \( h = -\frac{3}{2} \)
- \( k = 3.5 \)

### Step 7: Calculate \( k \)

Now, we need to find \( k \) in the equation \( x + y = k \):
\[ k = h + k = -\frac{3}{2} + \frac{7}{2} = \frac{4}{2} = 2 \]

Thus, the value of \( k \) is:
\[ \boxed{2} \]