The lengths of the tangents from any point on the circle `15x^(2)+15y^(2)-48x+64y=0` to the two circles `5x^(2)+5y^(2)-24x+32y+75=0` and `5x^(2)+5y^(2)-48x+64y+300=0` are in the ratio

Options

A. `1:2`
B. `2:3`
C. `3:4`
D. none of these

Answer

To solve the problem, we need to find the lengths of the tangents from a point on the first circle to the other two circles and then determine the ratio of these lengths.

### Step-by-Step Solution:

1. **Identify the First Circle:**
The equation of the first circle is given as:
\[ 15x^2 + 15y^2 - 48x + 64y = 0 \]
We can rewrite this in standard form by dividing the entire equation by 15:
\[ x^2 + y^2 - \frac{48}{15}x + \frac{64}{15}y = 0 \]
Completing the square for \(x\) and \(y\):
\[ \left(x - \frac{24}{15}\right)^2 + \left(y + \frac{32}{15}\right)^2 = \left(\frac{24^2 + 32^2}{15}\right) \]

2. **Identify the Second Circle:**
The second circle is given by:
\[ 5x^2 + 5y^2 - 24x + 32y + 75 = 0 \]
Dividing by 5:
\[ x^2 + y^2 - \frac{24}{5}x + \frac{32}{5}y + 15 = 0 \]
Completing the square:
\[ \left(x - \frac{12}{5}\right)^2 + \left(y + \frac{16}{5}\right)^2 = \left(\frac{12^2 + 16^2}{5} - 15\right) \]

3. **Identify the Third Circle:**
The third circle is given by:
\[ 5x^2 + 5y^2 - 48x + 64y + 300 = 0 \]
Dividing by 5:
\[ x^2 + y^2 - \frac{48}{5}x + \frac{64}{5}y + 60 = 0 \]
Completing the square:
\[ \left(x - \frac{24}{5}\right)^2 + \left(y + \frac{32}{5}\right)^2 = \left(\frac{24^2 + 32^2}{5} - 60\right) \]

4. **Length of Tangents:**
The length of the tangent from a point \(P(a, b)\) on the first circle to the second circle is given by:
\[ L_1 = \sqrt{S_1} = \sqrt{a^2 + b^2 - \frac{24}{5}a + \frac{32}{5}b + 15} \]
The length of the tangent to the third circle is:
\[ L_2 = \sqrt{S_2} = \sqrt{a^2 + b^2 - \frac{48}{5}a + \frac{64}{5}b + 60} \]

5. **Substituting \(a^2 + b^2\):**
From the first circle, we have:
\[ a^2 + b^2 = \frac{48}{15}a - \frac{64}{15}b \]
Substitute this into \(L_1\) and \(L_2\).

6. **Finding the Ratio \(L_1 : L_2\):**
After substituting and simplifying, we find that:
\[ L_2 = 2L_1 \]
Thus, the ratio of the lengths of the tangents is:
\[ L_1 : L_2 = 1 : 2 \]

### Final Answer:
The lengths of the tangents from any point on the first circle to the two circles are in the ratio \(1:2\).