The largest and shortest distance of the earth from the sun are r1 and r2 its distance from the

step 1 Here the correct answer is that is option C and now I will give here explanation So the explanat

The largest and shortest distance of the earth from the sun...

The largest and shortest distance of the earth from the sun are $\mathrm{r}_{1}$ and $\mathrm{r}_{2}$ its distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun (A) $\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 4\right]$ (B) $\left[\left(\mathrm{r}_{1} \mathrm{r}_{2}\right) /\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right)\right]$ (C) $\left[\left(2 r_{1} r_{2}\right) /\left(r_{1}+r_{2}\right)\right]$ (D) $\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 3\right]$

The largest and shortest distance of the earth from the sun are $\mathrm{r}_{1}$ and $\mathrm{r}_{2}$ its distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun
(A) $\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 4\right]$
(B) $\left[\left(\mathrm{r}_{1} \mathrm{r}_{2}\right) /\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right)\right]$
(C) $\left[\left(2 r_{1} r_{2}\right) /\left(r_{1}+r_{2}\right)\right]$
(D) $\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 3\right]$

Key Concepts

Ellipse Geometry

Ellipse geometry is a branch of conic sections that studies the shape defined as the locus of points for which the sum of the distances to two fixed points (the foci) is constant. It involves key properties such as the major and minor axes, the center, and the eccentricity, which dictates how “stretched” the ellipse is compared to a circle.

Perihelion and Aphelion Distances

In the context of planetary motion, the terms perihelion and aphelion refer to the closest and farthest points of the orbit of a planet around the sun, respectively. These distances can be expressed in terms of the semimajor axis and the eccentricity of the ellipse, with the perihelion distance given by a(1 ? e) and the aphelion distance by a(1 + e), directly linking the shape of the orbit to these key positions.

Relationship between Semi?major Axis, Eccentricity, and Focal Distances

The semimajor axis is a fundamental quantity in ellipse geometry, representing half the longest diameter of the ellipse, while the eccentricity quantifies its deviation from a circle. The positions of the foci are determined by the product ae, where a is the semimajor axis and e is the eccentricity. This relationship allows for derivations such as expressing the product of the perihelion and aphelion distances in terms of the semimajor axis and eccentricity.

Coordinate Analysis in Elliptical Orbits

Coordinate analysis involves setting up the ellipse with its equation and applying coordinate geometry to determine distances from the focus to points on the ellipse. For example, by considering a vertical line through a focus (which is perpendicular to the major axis) and using the standard equation of an ellipse, one can compute distances such as that from the focus to the ellipse along that perpendicular, highlighting the interplay between algebraic and geometric properties of the orbit.

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