Question

The angle between the tangents drawn from the origin to the circle \((x-7)^2+(y+1)^2 = 25\) is

• A. \(45^{\circ}\)
• B. \(90^{\circ}\)
• C. \(60^{\circ}\)
• D. \(30^{\circ}\)

Hint:

Use sin(half angle) = radius / distance of the point from the centre.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
From an external point at distance \(d\) from the centre of a circle of radius \(r\), the angle \(2\alpha\) between the two tangents satisfies \(\sin\alpha = r/d\).

Step 2: Compute
The circle has centre \((7, -1)\) and radius 5. Distance from the origin:
\[ d = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt2 \]
\[ \sin\alpha = \frac{5}{5\sqrt2} = \frac{1}{\sqrt2} \Rightarrow \alpha = 45^{\circ} \]

Step 3: Angle
The angle between the tangents is \(2\alpha = 90^{\circ}\). The origin lies outside the circle because \(d = 7.07 > 5\), so tangents exist.

Final Answer:
The angle is \(90^{\circ}\), option (B). \[ \boxed{90^{\circ}} \]