Information Given:
A children’s park has a circular garden as shown in the figure with DC and AB representing x and y-axis respectively. AB and CD are two cemented walkways of length 70 m each. Areas marked as AEOF and BGOH are flower beds whereas ECGO and FDHO are covered with grass. FOE and HOG are parabolas with focus (0, 6) and (0, −6) respectively.
Based on the information given above answer the questions.
If EFHG is another rectangular pathway in the circular garden, then area of EFHG is equal to:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(1000\sqrt{6}\) m2
The garden is the circle \(x^2+y^2 = 1225\) (radius 35), and the parabolas are \(x^2 = 24y\) (FOE) and \(x^2 = -24y\) (HOG). Putting \(x^2 = 24y\) in the circle:
\(y^2+24y-1225 = 0\)
\((y-25)(y+49) = 0,\quad y = 25\)
\(x^2 = 600,\quad x = \pm 10\sqrt{6}\)
So the corners of the rectangle are
\(E(10\sqrt{6}, 25),\ F(-10\sqrt{6}, 25)\)
\(H(-10\sqrt{6}, -25),\ G(10\sqrt{6}, -25)\)
Its sides are
\(EF = 10\sqrt{6}-(-10\sqrt{6}) = 20\sqrt{6}\)
\(EG = 25-(-25) = 50\)
Hence
\(\text{Area} = 20\sqrt{6} \times 50 = 1000\sqrt{6} \text{ m}^2\)