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Q.
An arc is in the form of a semi-ellipse. It is $8 \,m$ wide and $2 \,m$ high at the centre. Find the approximate height of the arc at a point $1.5 \,m$ from one end.
Conic Sections
Solution:
The semi-elliptical arc has been shown in the figure. We choose the coordinate axes as shown in the figure.
Here $OA = \frac{1}{2}\cdot$ $8 = 4$ and $OB = 2$
$\therefore $ The equation of the ellipse is
$\frac{x^{2}}{4^{2}}+\frac{y^{2}}{2^{2}}=1$ i.e., $x^{2} + 4y^{2} = 16\, \ldots\left(i\right)$
Let $h$ metres be the height of the arc at a distance of $1.5 \,m$ from one end, then $OM = 4 - 1.5 = 2.5$, so that the point $P$ is $\left(2.5, h\right)$.
Since $P$ lies on ellipse $\left(i\right)$
$\therefore \, \left(\frac{5}{2}\right)^{2}+4h^{2}=16$
$\Rightarrow \, 4h^{2}=16-\frac{25}{4}$
$\Rightarrow \, h=\frac{\sqrt{39}}{4}=1.56 \,m $ (approx.) $\left[\because\, h >0\right]$
