The point (1,2) lies inside and (3,4) outside the circle `x ^(2) +y ^(2) - 7x + 15 y - c =0,` if

Options

A. `c = 25`
B. `c = 35`
C. `c = 65`
D. c take any real value

Answer

To solve the problem, we need to determine the value of \( c \) for the circle given by the equation:

\[ x^2 + y^2 - 7x + 15y - c = 0 \]

We know that the point \( (1, 2) \) lies inside the circle and the point \( (3, 4) \) lies outside the circle. We will use the condition that relates the points to the circle to find the required values of \( c \).

### Step 1: Substitute the point \( (1, 2) \) into the circle equation

For the point \( (1, 2) \):

\[ S_1 = 1^2 + 2^2 - 7(1) + 15(2) - c \]

Calculating this:

\[ S_1 = 1 + 4 - 7 + 30 - c \]
\[ S_1 = 28 - c \]

Since the point \( (1, 2) \) lies inside the circle, we have:

\[ S_1 < 0 \implies 28 - c < 0 \implies c > 28 \]

### Step 2: Substitute the point \( (3, 4) \) into the circle equation

For the point \( (3, 4) \):

\[ S_2 = 3^2 + 4^2 - 7(3) + 15(4) - c \]

Calculating this:

\[ S_2 = 9 + 16 - 21 + 60 - c \]
\[ S_2 = 64 - c \]

Since the point \( (3, 4) \) lies outside the circle, we have:

\[ S_2 > 0 \implies 64 - c > 0 \implies c < 64 \]

### Step 3: Combine the inequalities

From Step 1, we found that \( c > 28 \) and from Step 2, we found that \( c < 64 \). Therefore, we can combine these results:

\[ 28 < c < 64 \]

### Conclusion

The value of \( c \) must lie in the interval:

\[ c \in (28, 64) \]