The angle between the tangents to the circle with centre (4,5) drawn from P(-2,-3) is `120^(@)` then length of the tagent to the circle from P is

Options

A. 4
B. 3
C. 2
D. 5

Answer

To find the length of the tangent to the circle from point P(-2, -3), we can follow these steps:

### Step 1: Identify the center of the circle and the point P
The center of the circle is given as O(4, 5) and the point P is given as P(-2, -3).

### Step 2: Calculate the distance OP
We will use the distance formula to find the distance between points O and P.

The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of O and P:
\[ OP = \sqrt{(4 - (-2))^2 + (5 - (-3))^2} \]
\[ = \sqrt{(4 + 2)^2 + (5 + 3)^2} \]
\[ = \sqrt{6^2 + 8^2} \]
\[ = \sqrt{36 + 64} \]
\[ = \sqrt{100} \]
\[ = 10 \]

### Step 3: Use the angle between the tangents
Given that the angle between the tangents (∠BPA) is 120°, we can find the angle ∠AOP. The angle ∠AOP is half of ∠BPA:
\[ \angle AOP = \frac{1}{2} \times 120° = 60° \]

### Step 4: Apply the cosine rule in triangle AOP
In triangle AOP, we can use the cosine of angle AOP to find the length of the tangent AP. The cosine rule states:
\[ \cos(\theta) = \frac{adjacent}{hypotenuse} \]
Here, we have:
\[ \cos(60°) = \frac{AP}{OP} \]
We know that:
\[ \cos(60°) = \frac{1}{2} \]
Substituting the known values:
\[ \frac{1}{2} = \frac{AP}{10} \]

### Step 5: Solve for AP
Multiplying both sides by 10:
\[ AP = 10 \times \frac{1}{2} = 5 \]

### Conclusion
The length of the tangent from point P to the circle is 5.