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.
The equation of diagonal EH of rectangular pathway EFHG is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(y=\displaystyle\frac{5}{2\sqrt{6}}x\)
The garden is the circle \(x^2+y^2 = 1225\) (radius 35), and FOE is the parabola \(x^2 = 24y\). They meet where
\(y^2+24y-1225 = 0,\quad y = 25,\quad x = \pm 10\sqrt{6}\)
So \(E(10\sqrt{6}, 25)\), and by symmetry of HOG \((x^2 = -24y)\), \(H(-10\sqrt{6}, -25)\).
E and H are opposite each other through the origin, so the diagonal EH passes through O. Its slope is
\( \displaystyle m = \frac{25-(-25)}{10\sqrt{6}-(-10\sqrt{6})} = \frac{50}{20\sqrt{6}} = \frac{5}{2\sqrt{6}} \)
Hence EH is
\( \displaystyle y = \frac{5}{2\sqrt{6}}\,x \)