Question

The parabolas \(y^{2}=4x\) and \(x^{2}=4y\) divide the square region bounded by the lines \(x=4,y=4\) and the coordinate axes. If \(A1,A2,A3\) are respectively the areas of these parts numbered from top to bottom, then \(A1:A2:A3\) is ...

• A. \(1:2:3\)
• B. \(3:2:1\)
• C. \(1:1:1\)
• D. \(1:1:2\)

Hint:

Find each of the three areas inside the square 0 to 4 and compare them.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The parabolas \(y^{2}=4x\) and \(x^{2}=4y\) meet at \((0,0)\) and \((4,4)\), cutting the square into three parts.

Step 2: Area below x^2 = 4y
\(A_{3}=\int _{0}^{4}\dfrac{x^{2}}{4}dx=\dfrac{16}{3}\).

Step 3: Area above y^2 = 4x
\(A_{1}=\int _{0}^{4}(4-2\sqrt{x})dx=16-\dfrac{32}{3}=\dfrac{16}{3}\).

Step 4: Middle part
\(A_{2}=16-\dfrac{16}{3}-\dfrac{16}{3}=\dfrac{16}{3}\).
So \(A_{1}:A_{2}:A_{3}=1:1:1\).

Final Answer:
The three areas are in the ratio \(1:1:1\). \[ \boxed{\text{(C) }1:1:1} \]