The largest and the shortest distance of the earth from the sun are `r_(1)` and `r_(2)`, its distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun

Options

A. `(r_(1)+r_(2))/4`
B. `(r_(1)r_(2))/(r_(1)+r_(2))`
C. `(2r_(1)r_(2))/(r_(1)+r_(2))`
D. `(r_(1)r_(2))/(3)`

Explanation

To solve the problem of finding the distance of the Earth from the Sun when it is at the perpendicular to the major axis of its elliptical orbit, we can follow these steps:

### Step 1: Understand the parameters
The largest distance of the Earth from the Sun is denoted as \( r_1 \) (aphelion), and the shortest distance is denoted as \( r_2 \) (perihelion). The semi-major axis \( a \) and the semi-minor axis \( b \) of the elliptical orbit are related to these distances.

### Step 2: Define the semi-major axis and eccentricity
The semi-major axis \( a \) can be defined as:
\[ a = \frac{r_1 + r_2}{2} \]
The eccentricity \( e \) of the ellipse is given by:
\[ e = \frac{r_1 - r_2}{r_1 + r_2} \]

### Step 3: Calculate the semi-minor axis
The semi-minor axis \( b \) can be calculated using the relationship:
\[ b = a \sqrt{1 - e^2} \]
Substituting the expression for \( e \):
\[ e^2 = \left(\frac{r_1 - r_2}{r_1 + r_2}\right)^2 \]
This gives:
\[ 1 - e^2 = 1 - \left(\frac{r_1 - r_2}{r_1 + r_2}\right)^2 \]

### Step 4: Find the distance when the Earth is perpendicular to the major axis
When the Earth is at the position perpendicular to the major axis, the distance \( d \) from the Sun can be calculated using the formula:
\[ d = \frac{2b^2}{a} \]
Substituting \( b^2 = a^2(1 - e^2) \):
\[ d = \frac{2a^2(1 - e^2)}{a} = 2a(1 - e^2) \]

### Step 5: Substitute the values of \( a \) and \( e^2 \)
Now substituting the values of \( a \) and \( e^2 \):
\[ d = 2 \left(\frac{r_1 + r_2}{2}\right) \left(1 - \left(\frac{r_1 - r_2}{r_1 + r_2}\right)^2\right) \]

### Step 6: Simplify the expression
After simplification, we can express \( d \) in terms of \( r_1 \) and \( r_2 \):
\[ d = \frac{2(r_1 + r_2)}{2} \left(1 - \frac{(r_1 - r_2)^2}{(r_1 + r_2)^2}\right) \]

### Final Expression
The final expression for the distance \( d \) when the Earth is at the perpendicular to the major axis is:
\[ d = \frac{2r_1r_2}{r_1 + r_2} \]