Question

The longest chord of the circumcircle of the triangle made by x-axis, y-axis and \( 4x+3y=24 \) is:

• A. 5 units
• B. 10 units
• C. 2.5 units
• D. 8 units

Hint:

For any right-angled triangle, the hypotenuse length is exactly equal to the diameter of its circumcircle.

Answer 1

The Correct Option is B

Step 1: Identify the Triangle:
The lines are:
1. \( y = 0 \) (x-axis)
2. \( x = 0 \) (y-axis)
3. \( 4x + 3y = 24 \)
Intercepts for the third line: If \( y=0, 4x=24 \implies x=6 \). Vertex A = \((6,0)\). If \( x=0, 3y=24 \implies y=8 \). Vertex B = \((0,8)\). Origin O = \((0,0)\). The triangle is a right-angled triangle with legs 6 and 8. Step 2: Circumcircle Properties:
For a right-angled triangle, the circumcenter lies at the midpoint of the hypotenuse, and the hypotenuse is the diameter of the circumcircle.
Length of hypotenuse = \( \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10 \). Step 3: Longest Chord:
The longest chord of any circle is its diameter. Since the hypotenuse is the diameter, the length is 10. Step 4: Final Answer:
10 units.