Two tangents are drawn from the origin to the circle $x^{2}+y^{2}-14x+2y+25=0$. Angle between these two tangents will be :

Question

Accepted Answer

To determine the angle between the two tangents drawn from the origin to the given circle, we first need to analyze the equation of the circle and the geometry involved.

The equation of the circle is given as:
$$ x^2 + y^2 - 14x + 2y + 25 = 0. $$

First, we rewrite this equation in the standard form of a circle $(x - h)^2 + (y - k)^2 = r^2$ by completing the square.

1. **Completing the square for $x$:**
   $$ x^2 - 14x $$
   $$ = (x - 7)^2 - 49. $$

2. **Completing the square for $y$:**
   $$ y^2 + 2y $$
   $$ = (y + 1)^2 - 1. $$

Substituting these into the original equation, we have:
$$ (x - 7)^2 - 49 + (y + 1)^2 - 1 + 25 = 0. $$

Simplifying, we get:
$$ (x - 7)^2 + (y + 1)^2 = 25. $$

This is the equation of a circle with center $(7, -1)$ and radius $5$.

To find the angle between the tangents from the origin to this circle, we use the formula for the angle $\theta$ between two tangents drawn from an external point to a circle:
$$ \cos \theta = \frac{d}{r}, $$
where $d$ is the distance from the external point to the center of the circle, and $r$ is the radius of the circle.

1. **Calculate $d$:** The distance from the origin $(0, 0)$ to the center $(7, -1)$ is:
   $$ d = \sqrt{(7 - 0)^2 + (-1 - 0)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}. $$

2. **Calculate $\cos \theta$:**
   $$ \cos \theta = \frac{d}{r} = \frac{5\sqrt{2}}{5} = \sqrt{2}. $$

However, since $\cos \theta$ cannot be greater than 1, this indicates that the tangents are perpendicular, and the angle between them is $90^\circ$.

Therefore, the angle between the two tangents is $90^\circ$, which corresponds to Option 4.

**Explanation for Incorrect Options:**

1. **Option 1: $0^\circ$** - This would imply that the tangents are coincident, which is not possible as the tangents are drawn from an external point to a circle.

2. **Option 2: $30^\circ$** - This angle would imply a specific cosine value that does not match the calculated $\cos \theta$.

3. **Option 3: $45^\circ$** - Similarly, this angle would imply a specific cosine value that does not match the calculated $\cos \theta$.

Thus, the correct answer is indeed Option 4, $90^\circ$, as the tangents are perpendicular to each other.