Question

Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L is

• A. 8
• B. 9
• C. 10
• D. 7

Answer 1

The Correct Option is B

A triangle is given where the equation of one side is \(3x + 5y - 45 = 0\). We need to determine the radius of the circumcircle of this triangle, which is a right-angled triangle. 

Concept Used:

In a right-angled triangle, the hypotenuse serves as the diameter of the circumcircle. Hence, the radius is: 
 

\[\text{Radius} = \frac{\text{Hypotenuse}}{2}\]

Step 1: Find Intercepts

The equation \(3x + 5y = 45\) gives the intercepts:

• Put \(x = 0\): \(5y = 45 \Rightarrow y = 9\)
• Put \(y = 0\): \(3x = 45 \Rightarrow x = 15\)

So, the triangle has vertices at: \((0, 0), (0, 9), (15, 0)\)

 

Step 2: Compute Hypotenuse

The hypotenuse is between the points \((0, 9)\) and \((15, 0)\): 
 

\[\text{Hypotenuse} = \sqrt{(15 - 0)^2 + (0 - 9)^2} = \sqrt{225 + 81} = \sqrt{306} \approx 17.5\]

Step 3: Compute Radius

\[\text{Radius} = \frac{17.5}{2} \approx 8.75\]

 Rounding to the nearest integer: \(\boxed{9}\)

Final Answer:

The required radius (to the nearest integer) is \(\boxed{9}\).