Question

If \( a, c, b \) are in GP, then the area of the triangle formed by the lines \( ax + by + c = 0 \) with the coordinate axes is equal to:

• A. 1
• B. 2
• C. \( \frac{1}{2} \)
• D. None of these

Hint:

When given a triangle formed by the coordinate axes and a line, you can use the formula for the area of a triangle to calculate it. If the coefficients of the line are in geometric progression, use the relationship between the coefficients to simplify the area expression.

Answer 1

The Correct Option is C

If \( a, c, b \) are in GP, then \( c^2 = ab \).
The area of the triangle formed by the line \( ax + by + c = 0 \) and the coordinate axes is calculated based on the intercepts. The x-intercept is \( x = \frac{-c}{a} \) and the y-intercept is \( y = \frac{-c}{b} \).The area of triangle \( AOB \) is given by:\[{Area} = \frac{1}{2} \times \left| \frac{-c}{a} \right| \times \left| \frac{-c}{b} \right| = \frac{1}{2} \times \frac{c^2}{ab}\]Since \( c^2 = ab \), substituting this into the area formula yields:\[{Area} = \frac{1}{2} \times \frac{c^2}{ab} = \frac{1}{2} \times \frac{ab}{ab} = \frac{1}{2}.\]Therefore, the area of the triangle is \( \frac{1}{2} \).