Question

The statement pattern \([(p∧q)\rightarrow (\sim p∨r)]∨[(\sim p∨r)\rightarrow (p∧q)]\) is

• A. a contradiction
• B. a tautology
• C. equivalent to \((p∧q)∨r\).
• D. equivalent to \((p∨q)\) .

Hint:

Find the length of tangent and radius, then use the half angle sine.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The circle has centre \(C(7, -1)\) and radius 5. The distance from the origin O to the centre is \(OC = \sqrt{49 + 1} = \sqrt{50}\).

Step 2: Half angle:
If \(2\alpha\) is the angle between the tangents, then in the right triangle formed by O, C and the point of contact, \(\sin\alpha = \frac{r}{OC} = \frac{5}{\sqrt{50}} = \frac{1}{\sqrt2}\). So \(\alpha = 45^\circ\).

Step 3: Angle between tangents:
\(2\alpha = 90^\circ\). This also agrees with the power of the origin: \(S_{11} = 49 + 1 - 25 = 25 = r^2\), the condition for perpendicular tangents.

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