Question

A straight line passes through the point \((3,3)\) and intersects the positive \(x\) and \(y\) axes at the points \(A(h,0)\) and \(B(0,k)\) respectively. If O is the origin, then the minimum area of \(\triangle OAB\) is:

• A. \(9\) square units.
• B. \(18\) square units.
• C. \(27\) square units.
• D. \(36\) square units.

Hint:

Use the intercept form and apply AM-GM to the condition 3/h + 3/k = 1.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The line is \(\dfrac{x}{h}+\dfrac{y}{k}=1\) with \(h,k>0\).

Step 2: Use the Point
Passing through \((3,3)\): \(\dfrac{3}{h}+\dfrac{3}{k}=1\).

Step 3: Apply AM-GM
\(1=\dfrac{3}{h}+\dfrac{3}{k}\ge2\sqrt{\dfrac{9}{hk}}\Rightarrow\sqrt{hk}\ge6\Rightarrow hk\ge36\).

Step 4: Minimum Area
Area \(=\dfrac{1}{2}hk\ge18\), option (B).

Final Answer:
The minimum area is 18 square units. \[ \boxed{\text{(B) }18\ \text{square units}} \]