Let C be the circle concentric with the circle , `2x^(2) + 2y^(2) - 6x - 10 y = 183 ` and having area `((1)/(10))^(th)` of the area of this circle. Then a tangent to C, parallel to the line, 3x + y = 0 makes an intercept on the y-axis , which is equal to :

Options

A. `-10`
B. `-4`
C. 17
D. 14

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Find the equation of the given circle
The equation of the circle is given as:

\[ 2x^2 + 2y^2 - 6x - 10y = 183 \]

We can simplify this by dividing the entire equation by 2:

\[ x^2 + y^2 - 3x - 5y = \frac{183}{2} \]

### Step 2: Rewrite the equation in standard form
To rewrite the equation in standard form, we need to complete the square for both \(x\) and \(y\).

1. For \(x\):
\[ x^2 - 3x \rightarrow \left(x - \frac{3}{2}\right)^2 - \frac{9}{4} \]

2. For \(y\):
\[ y^2 - 5y \rightarrow \left(y - \frac{5}{2}\right)^2 - \frac{25}{4} \]

Putting it all together:

\[ \left(x - \frac{3}{2}\right)^2 - \frac{9}{4} + \left(y - \frac{5}{2}\right)^2 - \frac{25}{4} = \frac{183}{2} \]

Combine the constants:

\[ \left(x - \frac{3}{2}\right)^2 + \left(y - \frac{5}{2}\right)^2 = \frac{183}{2} + \frac{9}{4} + \frac{25}{4} \]

Finding a common denominator (4):

\[ \frac{366}{4} + \frac{9}{4} + \frac{25}{4} = \frac{400}{4} = 100 \]

Thus, the equation of the circle becomes:

\[ \left(x - \frac{3}{2}\right)^2 + \left(y - \frac{5}{2}\right)^2 = 100 \]

### Step 3: Determine the radius of the circle
From the standard form, we see that the radius \(R\) of the circle is:

\[ R = \sqrt{100} = 10 \]

### Step 4: Find the area of the circle
The area \(A\) of the circle is given by:

\[ A = \pi R^2 = \pi (10^2) = 100\pi \]

### Step 5: Find the area of circle \(C\)
The area of circle \(C\) is given as \(\frac{1}{10}\) of the area of the original circle:

\[ \text{Area of circle } C = \frac{1}{10} \times 100\pi = 10\pi \]

### Step 6: Determine the radius of circle \(C\)
Let the radius of circle \(C\) be \(r\). Then:

\[ \pi r^2 = 10\pi \]

Dividing by \(\pi\):

\[ r^2 = 10 \implies r = \sqrt{10} \]

### Step 7: Find the equation of the tangent to circle \(C\)
The slope of the line \(3x + y = 0\) is:

\[ m = -3 \]

The equation of the tangent to circle \(C\) can be expressed as:

\[ y = mx + c \]

Since it is parallel to the line, we have:

\[ y = -3x + c \]

### Step 8: Use the distance formula to find \(c\)
The distance from the center of circle \(C\) (which is at \((\frac{3}{2}, \frac{5}{2})\)) to the line must equal the radius \(\sqrt{10}\).

Using the formula for the distance from a point to a line \(Ax + By + C = 0\):

\[ \text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

Here, \(A = 3\), \(B = 1\), and \(C = -c\). The distance is:

\[ \frac{|3(\frac{3}{2}) + 1(\frac{5}{2}) - c|}{\sqrt{3^2 + 1^2}} = \sqrt{10} \]

Calculating the numerator:

\[ | \frac{9}{2} + \frac{5}{2} - c | = | \frac{14}{2} - c | = |7 - c| \]

Thus, we have:

\[ \frac{|7 - c|}{\sqrt{10}} = \sqrt{10} \]

Multiplying both sides by \(\sqrt{10}\):

\[ |7 - c| = 10 \]

### Step 9: Solve for \(c\)
This gives us two cases:

1. \(7 - c = 10 \implies c = -3\)
2. \(7 - c = -10 \implies c = 17\)

### Step 10: Find the y-intercepts
1. For \(c = -3\):
\[ y = -3x - 3 \implies y\text{-intercept} = -3 \]

2. For \(c = 17\):
\[ y = -3x + 17 \implies y\text{-intercept} = 17 \]

### Conclusion
The intercepts on the y-axis are \(-3\) and \(17\). The problem asks for the intercept which is equal to:

\[ \text{Answer: } 17 \]