Question

An arch is in the form of semi-ellipse. It is $8m$ wide and $2m$ high at the centre. Then, the height of the arch at a point $1.5m$ from one end is

• A. 1.56 m
• B. 2.4375 m
• C. 2.056 m
• D. 1.086 m

Answer 1

Answer: (A)

Clearly, equation of ellipse takes the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1...(i)$
Here, it is given $2a=8$ and $b=2\Rightarrow a=4,b=2$
Put the values of a and b in eq. (i), we get
$\frac{x^2}{16}+\frac{y^2}{4}=1$

Given, $AP=1.5m\Rightarrow OP=OA-AP=4-1.5$
$\Rightarrow OP=2.5M$
Let $PQ=k$
$\therefore$ Coordinate $Q=(2.5,k)$ will satisfy the equation of ellipse.
i.e.,$\frac{(2.5{)}^2}{16}+\frac{k^2}{4}=1\Rightarrow \frac{6.25}{16}+\frac{k^2}{4}=1$
$\Rightarrow \frac{k^2}{4}=\frac{1}{1}-\frac{6.25}{16}=\frac{16-6.25}{16}$
$\frac{k^2}{4}=\frac{9.75}{16}\Rightarrow k^2=\frac{9.75}{4}$
$\Rightarrow k^2=2.4375\Rightarrow k=1.56m($ approx. $)$