Question

The satellite is moving in an elliptical orbit about the earth as shown in figure:


The minimum and maximum distances of satellite from earth are \(3~\text{units}\) and \(5~\text{ units}\), respectively. The distance of satellite from the earth when it is at \(\text{P}\) is equal to

• A. $3\text{units}$
• B. None of thse
• C. $4\text{units}$
• D. $3.75\text{units}$

Answer 1

The correct option is C $4\text{units}$


From the figure,
$r_1=(1-e)a=3....\left(1\right)$
$r_2=(1+e)a=5....\left(2\right)$

From equation $\left(1\right)$ and $\left(2\right)$,

$r_1+r_2=2a=8$

$\Rightarrow a=4$

Then, $\text{OE}=\text{OA - AE}=4-3=1$

For an ellipse $(a>b)$,

$\text{OE}=ae=1$

$\Rightarrow e=\frac{1}{a}=\frac{1}{4}$

Semi-minor axis , $b=a\sqrt{1-e^2}=4\sqrt{1-\frac{1}{16}}=\sqrt{15}$

Required distance, $\text{EP}=\sqrt{{\left(OP\right)}^2+{\left(OE\right)}^2}$

$\Rightarrow EP=\sqrt{b^2+{\left(ae\right)}^2}=\sqrt{15+1}$

$\therefore \text{EP}=4\text{units}$

Hence, option (a) is the correct answer.

Why this question? To make students familiar with the properties of an ellipse so that they can handle problems on elliptical orbits.