Finding the Angle Between Tangents to a Circle

The problem asks for the angle between the tangents drawn from a specific point to a given circle. First, we need to determine the coordinates of the point after translation and then use the properties of tangents to a circle.

1. Determine the Translated Point Coordinates

The initial point is $P_0 = (1, 2)$.

It is translated 2 units in the positive direction of the x-axis.

The new point, let's call it $P'$, has coordinates:

$P' = (1 + 2, 2) = (3, 2)$

2. Identify Circle Properties

The equation of the circle is given as $x^2 + y^2 = 9$.

• The center of the circle is $C = (0, 0)$.
• The radius of the circle is $r = \sqrt{9} = 3$.

3. Calculate the Distance from the Point to the Circle's Center

We need to find the distance between the translated point $P'(3, 2)$ and the center $C(0, 0)$.

Using the distance formula, $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$:

$CP' = \sqrt{(3 - 0)^2 + (2 - 0)^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$

4. Calculate the Angle Between Tangents

Let the angle between the two tangents drawn from $P'$ to the circle be $2\theta$.

Consider the right-angled triangle formed by the center $C$, the point $P'$, and one of the tangent points $T$. The angle $\angle CP'T = \theta$.

In this triangle, $CT$ is the radius ($r=3$) and $CP'$ is the distance ($\sqrt{13}$).

We can use trigonometry to find $\theta$. Specifically, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{CT}{CP'}$:

$\sin(\theta) = \frac{3}{\sqrt{13}}$

Alternatively, we can find the length of the tangent segment $P'T$. Using the Pythagorean theorem in $\triangle CP'T$:

$CP'^2 = CT^2 + P'T^2$

$(\sqrt{13})^2 = 3^2 + P'T^2$

$13 = 9 + P'T^2$

$P'T^2 = 13 - 9 = 4$

$P'T = \sqrt{4} = 2$

Now, we can find $\tan(\theta)$:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CT}{P'T} = \frac{3}{2}$

Therefore, $\theta = \tan^{-1}\left(\frac{3}{2}\right)$.

The angle between the tangents is $2\theta$.

$2\theta = 2\tan^{-1}\left(\frac{3}{2}\right)$

Conclusion

The angle between the tangents drawn from the translated point $(3, 2)$ to the circle $x^2 + y^2 = 9$ is $2\tan^{-1}\left(\frac{3}{2}\right)$.