Question

An arch way is in the slope of a semi-ellipse, the road level being the major axis. If the breadth of the road is 30 m and the height of the arch is 6 m at a distance of 2 m from the side, the greatest height of the arch is–

• A. 6 m
• B. 12 m
• C. 10 m
• D. 5.5m

Answer 1

Answer: (A)

6 m

Answer 2

Equation of the ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}$= 1

The point P ( – (a – 2), 6) lies on it

$\frac{(a - 2)^{2}}{a^{2}}$ + $\frac{36}{b^{2}}$ = 1

$\frac{36}{b^{2}}$= 1 – $\frac{(a - 2)^{2}}{a^{2}}$ ̃ $\frac{4(a - 1)}{a^{2}}$

$\frac{9}{b^{2}}$ = $\frac{a - 1}{a^{2}}$

b² = $\frac{9a^{2}}{a - 1}$ ̃ $9\left\{ a + 1 + \frac{1}{a - 1} \right\}$

$\frac{d(b^{2})}{da}$ = 9$\left\{ 1 - \frac{1}{(a - 1)^{2}} \right\}$ and $\frac{d^{2}b^{2}}{da^{2}}$ = $\frac{18}{(a - 1)^{3}}$

For extreme value of b

$\frac{db^{2}}{da}$= 0 ̃ a = 2

b² = $\frac{9(2)^{2}}{2 - 1}$ ̃ b = 6m

greatest height of the arch is 6m.