Question

Area of a triangle formed by the line \(x \cos \alpha + y \sin \alpha = p\) with the coordinate axes is

• A. \(\frac{p^2}{2 \sin \alpha \cos \alpha}\)
• B. \(\frac{p^2}{ \sin \alpha \cos \alpha}\)
• C. \(\frac{p}{2 \sin \alpha \cos \alpha}\)
• D. \(\frac{p}{ \sin \alpha \cos \alpha}\)

Answer 1

The Correct Option is A

To solve the problem, we need to find the area of the triangle formed by the line \( x \cos \alpha + y \sin \alpha = p \) with the coordinate axes.

1. Finding the Intercepts:
To find the x-intercept, set \( y = 0 \):
\[ x \cos \alpha = p \Rightarrow x = \frac{p}{\cos \alpha} \]
To find the y-intercept, set \( x = 0 \):
\[ y \sin \alpha = p \Rightarrow y = \frac{p}{\sin \alpha} \]

2. Using the Formula for Area of Triangle:
The area of a triangle formed by a line intersecting the coordinate axes at \( x = a \) and \( y = b \) is:
\[ \text{Area} = \frac{1}{2} \times a \times b \]
Substituting the intercepts:
\[ \text{Area} = \frac{1}{2} \times \frac{p}{\cos \alpha} \times \frac{p}{\sin \alpha} = \frac{p^2}{2 \sin \alpha \cos \alpha} \]

Final Answer:
The area of the triangle is \( \frac{p^2}{2 \sin \alpha \cos \alpha} \) (Option A).