Question

A bridge is in the shape of a semi-ellipse . It is $400$ metres long and has a maximum height of $10$ metres at the middle point. The height of the bridge at a point at a distance of $80$ metres from one end is

• A. $4$ metres
• B. $2$ metres
• C. $8$ metres
• D. $\sqrt{91}$ metres

Answer 1

The correct option is C $8$ metres

We can see that,

$AC=400\Rightarrow OA=\frac{400}{2}=200$

and

$OB=10$

Hence, $a=200;b=10$

So, Equation of Ellipse will be $\frac{x^2}{{200}^2}+\frac{y^2}{{10}^2}=1$

Now, at point P from end A $=AP=8$

$\Rightarrow OP=OA-PA=200-80=120$

Now, let point Q be $(120,y)$

Putting this in Equatio $\frac{x^2}{{200}^2}+\frac{y^2}{{10}^2}=1$

$\frac{{120}^2}{{200}^2}+\frac{y^2}{{10}^2}=1$

$\Rightarrow \frac{9}{25}+\frac{y^2}{100}=1\Rightarrow y^2=100\times \frac{16}{25}\Rightarrow y^2=64$

$\Rightarrow y=8$