PA and PB are two tangents drawn from point P to circle of radius 5 . A line is drawn from point P which cuts at C and D such that PC=5 and PD=15 and ` angleAPB= theta `.
On the basis of above information answer the questions .
Number of solution(s) of the equation `log_(cos theta )(x+2)=2+3 log_((x+2)) sin""((5 theta )/(2))` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to analyze the given equation step by step. The equation we need to solve is:

\[ \log_{\cos \theta}(x + 2) = 2 + 3 \log_{(x + 2)} \sin\left(\frac{5\theta}{2}\right) \]

### Step 1: Find the value of \(\theta\)

From the problem, we know that \(P\) is a point from which two tangents \(PA\) and \(PB\) are drawn to a circle of radius 5. The distances \(PC\) and \(PD\) are given as 5 and 15 respectively.

Using the Pythagorean theorem in triangle \(OAP\) (where \(O\) is the center of the circle), we have:

\[ OP^2 = OA^2 + AP^2 \]

Given that \(OP = PC + OC = 5 + 5 = 10\) and \(OA = 5\):

\[ 10^2 = 5^2 + AP^2 \implies 100 = 25 + AP^2 \implies AP^2 = 75 \implies AP = 5\sqrt{3} \]

Now, using the cosine of half the angle \(\theta\):

\[ \cos\left(\frac{\theta}{2}\right) = \frac{AP}{OP} = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2} \]

Thus,

\[ \frac{\theta}{2} = \frac{\pi}{6} \implies \theta = \frac{\pi}{3} \]

### Step 2: Substitute \(\theta\) into the equation

Now substituting \(\theta = \frac{\pi}{3}\) into the equation:

\[ \log_{\cos\left(\frac{\pi}{3}\right)}(x + 2) = 2 + 3 \log_{(x + 2)} \sin\left(\frac{5\pi}{6}\right) \]

Calculating \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}\):

The equation becomes:

\[ \log_{\frac{1}{2}}(x + 2) = 2 + 3 \log_{(x + 2)}\left(\frac{1}{2}\right) \]

### Step 3: Simplify the equation

Using the change of base formula for logarithms:

\[ \log_{\frac{1}{2}}(x + 2) = -\log_{(x + 2)}(x + 2) = -1 \]

Thus, we can rewrite the equation as:

\[ -\log_{(x + 2)}(x + 2) = 2 + 3(-1) \]

This simplifies to:

\[ -\log_{(x + 2)}(x + 2) = 2 - 3 = -1 \]

### Step 4: Solve for \(x\)

Now we have:

\[ \log_{(x + 2)}(x + 2) = 1 \]

This implies:

\[ x + 2 = (x + 2)^1 \implies x + 2 = 2 \implies x = 0 \]

### Step 5: Check for other solutions

Now, we also need to consider the case when:

\[ \log_{(x + 2)}\left(\frac{1}{2}\right) = -1 \]

This gives:

\[ x + 2 = \frac{1}{2} \implies x = -\frac{3}{2} \]

### Conclusion

Thus, we have two solutions:

1. \(x = 0\)
2. \(x = -\frac{3}{2}\)

Therefore, the number of solutions to the equation is **2**.