If `5 cos theta-12sintheta=0,` the value of `overset(2sintheta+costheta) (costheta-sintheta)` is:

Options

A. `1""_(119)^(75)`
B. `3""_(7)^(1)`
C. `2""_(35)^(34)`
D. `3""_(3)^(2)`

Answer

To solve the equation \(5 \cos \theta - 12 \sin \theta = 0\) and find the value of \(\frac{2 \sin \theta + \cos \theta}{\cos \theta - \sin \theta}\), we can follow these steps:

### Step 1: Solve for \(\tan \theta\)

Starting with the equation:

\[ 5 \cos \theta - 12 \sin \theta = 0 \]

We can rearrange this to find the ratio of \(\sin \theta\) and \(\cos \theta\):

\[ 5 \cos \theta = 12 \sin \theta \]

Dividing both sides by \(\cos \theta\) (assuming \(\cos \theta \neq 0\)) gives:

\[ 5 = 12 \tan \theta \]

Thus, we have:

\[ \tan \theta = \frac{5}{12} \]

### Step 2: Determine \(\sin \theta\) and \(\cos \theta\)

Using the Pythagorean identity, we can find \(\sin \theta\) and \(\cos \theta\). We can visualize this as a right triangle where the opposite side is 5 and the adjacent side is 12. The hypotenuse \(h\) can be calculated as:

\[ h = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

Now we can find \(\sin \theta\) and \(\cos \theta\):

\[ \sin \theta = \frac{5}{13}, \quad \cos \theta = \frac{12}{13} \]

### Step 3: Substitute into the expression

Now we substitute \(\sin \theta\) and \(\cos \theta\) into the expression \(\frac{2 \sin \theta + \cos \theta}{\cos \theta - \sin \theta}\):

First, calculate \(2 \sin \theta + \cos \theta\):

\[ 2 \sin \theta + \cos \theta = 2 \left(\frac{5}{13}\right) + \frac{12}{13} = \frac{10}{13} + \frac{12}{13} = \frac{22}{13} \]

Next, calculate \(\cos \theta - \sin \theta\):

\[ \cos \theta - \sin \theta = \frac{12}{13} - \frac{5}{13} = \frac{7}{13} \]

### Step 4: Final calculation

Now we can substitute these values back into the expression:

\[ \frac{2 \sin \theta + \cos \theta}{\cos \theta - \sin \theta} = \frac{\frac{22}{13}}{\frac{7}{13}} = \frac{22}{7} \]

### Conclusion

Thus, the final value is:

\[ \frac{22}{7} \]