Question

A straight line meets the coordinate axes at $A$ and $B$. A circle is circumscribed about the triangle $OAB$, with $O$ being the origin. If $m$ and $n$ are the distances of the tangent from the origin to the points $A$ and $B$ respectively, the diameter of the circle is

• A. m(m + n)
• B. m + n
• C. n(m + n)
• D. 1/2 ( m+n)

Answer 1

The Correct Option is B

Let the equation of the line be \( \frac{x}{a} + \frac{y}{b} = 1 \), where \( A(a, 0) \) and \( B(0, b) \) are the points of intersection of the line with the coordinate axes. The radius of the circle circumscribed about the triangle \( OAB \) is given by the formula: \[ R = \frac{abc}{4A} \] where \( A \) is the area of the triangle and \( abc \) is the product of the lengths of the sides of the triangle. However, we can simplify the situation using the concept of tangents.

Step 1: Tangent formula and relationship between the points
We are given that \( m \) and \( n \) are the distances of the tangents from the origin to the points \( A \) and \( B \), respectively. The formula for the length of the tangent from the origin to a point \( (x_1, y_1) \) is given by: \[ \text{Tangent length} = \sqrt{x_1^2 + y_1^2} \] Thus, for points \( A(a, 0) \) and \( B(0, b) \), the tangent lengths are: \[ m = \sqrt{a^2} = a, \quad n = \sqrt{b^2} = b \]

 Step 2: Diameter of the circumcircle
The formula for the diameter of the circumcircle is related to the sum of the distances from the origin to the points of tangency. The diameter \( D \) is simply the sum of the tangent lengths: \[ D = m + n \]

Answer:

\[ \boxed{m + n} \]