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:
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:
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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.
Updated On: Mar 26, 2025
- 1
- 2
- \( \frac{1}{2} \)
- None of these
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The Correct Option is C
Solution and Explanation
Given \( a, c, b \) are in GP, so \( c^2 = ab \).
The area of the triangle formed by the line \( ax + by + c = 0 \) and the coordinate axes can be found using the formula for the area of a triangle formed by two lines intersecting the axes at \( x = \frac{-c}{a} \) and \( y = \frac{-c}{b} \). The area of the triangle \( AOB \) is: \[ {Area} = \frac{1}{2} \times \left( \frac{-c}{b} \right) \times \left( \frac{-c}{a} \right) = \frac{1}{2} \times \frac{c^2}{ab} = \frac{1}{2} \times \frac{c^2}{ab} = \frac{1}{2} { (using } c^2 = ab). \] Thus, the area is \( \frac{1}{2} \).
The area of the triangle formed by the line \( ax + by + c = 0 \) and the coordinate axes can be found using the formula for the area of a triangle formed by two lines intersecting the axes at \( x = \frac{-c}{a} \) and \( y = \frac{-c}{b} \). The area of the triangle \( AOB \) is: \[ {Area} = \frac{1}{2} \times \left( \frac{-c}{b} \right) \times \left( \frac{-c}{a} \right) = \frac{1}{2} \times \frac{c^2}{ab} = \frac{1}{2} \times \frac{c^2}{ab} = \frac{1}{2} { (using } c^2 = ab). \] Thus, the area is \( \frac{1}{2} \).
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