Question

If the acute angle between the pair of tangents drawn from the origin to the circle \( x^2 + y^2 - 4x - 8y + 4 = 0 \) is \( \alpha \), then \( \tan \alpha = \)

• A. \( \frac{3}{5} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{4}{3} \)
• D. \( \frac{4}{5} \)

Hint:

Use the formula for the angle between the tangents drawn from a point to the circle, which involves the radius and center of the circle.

Answer 1

The Correct Option is C

Step 1: Rewrite the equation of the circle.
The given equation is \( x^2 + y^2 - 4x - 8y + 4 = 0 \). Completing the square, we get: \[ (x - 2)^2 + (y - 4)^2 = 16 \] So, the center of the circle is \( (2, 4) \) and the radius is \( r = 4 \).
Step 2: Use the formula for the angle between tangents.
The formula is: \[ \tan \alpha = \frac{r}{\sqrt{h^2 + k^2 - r^2}} \]
Step 3: Substitute the values.
\[ \tan \alpha = \frac{4}{\sqrt{2^2 + 4^2 - 4^2}} = \frac{4}{\sqrt{4}} = 2 \] Final Answer: \[ \boxed{2} \]