If the area of the quadrilateral by the tangents from the origin to the circle `x^2+y^2+6x-10 y+c=0` and the radii corresponding to the points of contact is `15 ,` then a value of `c` is 9 (b) 4 (c) 5 (d) 25

Options

A. 9
B. 4
C. 5
D. 25

Explanation

To find the value of \( c \) for the given circle equation \( x^2 + y^2 + 6x - 10y + c = 0 \) such that the area of the quadrilateral formed by the tangents from the origin and the radii to the points of contact is 15, we will follow these steps:

### Step 1: Rewrite the Circle Equation
We start with the circle's equation:
\[ x^2 + y^2 + 6x - 10y + c = 0 \]
We will complete the square for both \( x \) and \( y \).

### Step 2: Completing the Square
For \( x \):
\[ x^2 + 6x = (x + 3)^2 - 9 \]
For \( y \):
\[ y^2 - 10y = (y - 5)^2 - 25 \]
Substituting these back into the equation gives:
\[ (x + 3)^2 - 9 + (y - 5)^2 - 25 + c = 0 \]
Simplifying this:
\[ (x + 3)^2 + (y - 5)^2 + c - 34 = 0 \]
Thus, we have:
\[ (x + 3)^2 + (y - 5)^2 = 34 - c \]

### Step 3: Identify the Center and Radius
From the completed square form, we identify:
- Center: \( (-3, 5) \)
- Radius: \( r = \sqrt{34 - c} \)

### Step 4: Area of the Quadrilateral
The area \( A \) of the quadrilateral formed by the tangents from the origin and the radii is given as 15. The area can be expressed as:
\[ A = 2 \times \text{Area of triangle formed by the origin and points of tangency} \]
The area of the triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{(Length of tangent)} \times \text{(Radius)} \]
The length of the tangent from the origin to the circle is given by:
\[ \text{Length of tangent} = \sqrt{S_1} = \sqrt{c} \]
Thus, the area of the triangle becomes:
\[ \text{Area} = \frac{1}{2} \times \sqrt{c} \times \sqrt{34 - c} \]

### Step 5: Set Up the Equation
The area of the quadrilateral is:
\[ 2 \times \left( \frac{1}{2} \times \sqrt{c} \times \sqrt{34 - c} \right) = \sqrt{c} \sqrt{34 - c} \]
Setting this equal to 15:
\[ \sqrt{c} \sqrt{34 - c} = 15 \]

### Step 6: Square Both Sides
Squaring both sides gives:
\[ c(34 - c) = 225 \]
Expanding this:
\[ 34c - c^2 = 225 \]
Rearranging:
\[ c^2 - 34c + 225 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula:
\[ c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1, b = -34, c = 225 \):
\[ c = \frac{34 \pm \sqrt{(-34)^2 - 4 \cdot 1 \cdot 225}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 34^2 = 1156, \quad 4 \cdot 225 = 900 \]
Thus:
\[ c = \frac{34 \pm \sqrt{1156 - 900}}{2} = \frac{34 \pm \sqrt{256}}{2} = \frac{34 \pm 16}{2} \]
Calculating the two possible values:
1. \( c = \frac{50}{2} = 25 \)
2. \( c = \frac{18}{2} = 9 \)

### Step 8: Conclusion
The possible values of \( c \) are \( 25 \) and \( 9 \).

### Final Answer
The values of \( c \) are \( 9 \) and \( 25 \).