Tangents drawn from the point (4, 3) to the circle `x^(2)+y^(2)-2x-4y=0` are inclined at an angle

Options

A. `pi//6`
B. `pi//4`
C. `pi//3`
D. `pi//2`

Explanation

To solve the problem of finding the angle between the tangents drawn from the point (4, 3) to the circle given by the equation \( x^2 + y^2 - 2x - 4y = 0 \), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we need to rewrite the given circle equation in standard form. The equation is:

\[ x^2 + y^2 - 2x - 4y = 0 \]

We can rearrange it as:

\[ x^2 - 2x + y^2 - 4y = 0 \]

### Step 2: Complete the Square
Next, we complete the square for both \(x\) and \(y\):

For \(x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]

For \(y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]

Substituting these back into the equation gives:

\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 = 0 \]
\[ (x - 1)^2 + (y - 2)^2 - 5 = 0 \]
\[ (x - 1)^2 + (y - 2)^2 = 5 \]

### Step 3: Identify the Center and Radius
From the standard form of the circle \((x - h)^2 + (y - k)^2 = r^2\), we can identify:

- Center \(C(1, 2)\)
- Radius \(r = \sqrt{5}\)

### Step 4: Calculate the Distance from the Point to the Center
Now, we need to find the distance \(d\) from the point \(P(4, 3)\) to the center \(C(1, 2)\):

Using the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ d = \sqrt{(4 - 1)^2 + (3 - 2)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \]

### Step 5: Use the Angle Between the Tangents Formula
The angle \(\theta\) between the tangents can be found using the sine formula:

\[ \sin \theta = \frac{r}{d} \]

Substituting the known values:

\[ \sin \theta = \frac{\sqrt{5}}{\sqrt{10}} = \frac{1}{\sqrt{2}} \]

### Step 6: Find the Angle \(\theta\)
From \(\sin \theta = \frac{1}{\sqrt{2}}\), we find:

\[ \theta = 45^\circ \]

### Step 7: Calculate the Angle Between the Tangents
The angle between the two tangents is \(2\theta\):

\[ \text{Angle between tangents} = 2 \times 45^\circ = 90^\circ \]

In radians, this is:

\[ \frac{\pi}{2} \]

### Final Answer
Thus, the angle between the tangents drawn from the point (4, 3) to the circle is:

\[ \frac{\pi}{2} \]