6.4 Q-6

Question Statement

A comet has a parabolic orbit with the Earth at its focus. When the comet is $150,000 , \mathrm{km}$ from the Earth, the line joining the comet and the Earth makes an angle of $30^\circ$ with the axis of the parabola. Determine the closest distance the comet will come to the Earth.

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Background and Explanation

In this problem, the Earth is at the focus of the parabola, and the comet follows a parabolic trajectory. The given information about the distance ($150,000 , \mathrm{km}$) and angle ($30^\circ$) helps us relate the focus and vertex properties of the parabola.

Key concepts:

1. Parabola Definition: The distance from any point on the parabola to the focus equals its perpendicular distance to the directrix.
2. Coordinate Geometry: The relationship between the angle and distances in the parabola can be derived using trigonometric principles and the parabola’s standard equation.

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Solution

Step 1: Define the Problem

Let the Earth ($E$) be the focus of the parabola, placed at the origin $(0, 0)$.
The vertex of the parabola is at $(-a, 0)$, and the directrix is $x = -2a$.
The comet is at point $P(x, y)$, satisfying:

$$\sqrt{x^2 + y^2} = |x + 2a| \tag{1}$$

The distance from the comet to the Earth is given as:

$$x^2 + y^2 = (150,000)^2 \tag{2}$$

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Step 2: Relating $x$ and $y$ Using Geometry

From the triangle formed by the Earth ($E$), the comet ($P$), and the axis of the parabola, the angle between the line $EP$ and the axis is $30^\circ$. Using trigonometry:

$$\cos 30^\circ = \frac{x}{150,000} \implies x = 150,000 \cdot \cos 30^\circ$$

Substituting $\cos 30^\circ = \frac{\sqrt{3}}{2}$:

$$x = 150,000 \cdot \frac{\sqrt{3}}{2} = 75,000 \cdot \sqrt{3} \tag{3}$$

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Step 3: Solve for the Distance $a$

Using equation (1) and substituting $x = 75,000 \cdot \sqrt{3}$:

$$\sqrt{x^2 + y^2} = x + 2a$$

From equation (2):

$$x^2 + y^2 = (150,000)^2$$

Thus:

$$(75,000 \cdot \sqrt{3})^2 + y^2 = (150,000)^2 \implies y^2 = (150,000)^2 - (75,000 \cdot \sqrt{3})^2$$

Substitute $x + 2a = 150,000$:

$$75,000 \cdot \sqrt{3} + 2a = 150,000$$

Rearranging:

$$2a = 150,000 - 75,000 \cdot \sqrt{3}$$

Factorize:

$$2a = 75,000(2 - \sqrt{3})$$

Solve for $a$:

$$a = 75,000 \cdot \frac{2 - \sqrt{3}}{2} \approx 10,048.095 , \mathrm{km}$$

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Step 4: Closest Distance to the Earth

The closest distance of the comet to the Earth is the vertex of the parabola, which is $a$. Thus:

$$\text{Closest Distance} = 75,000 \cdot \frac{2 - \sqrt{3}}{2} \approx 10,048.095 , \mathrm{km}$$

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Key Formulas or Methods Used

1. Definition of a Parabola:

$$\sqrt{x^2 + y^2} = |x + 2a|$$

2. Trigonometric Relation:

$$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$

3. Distance Formula:

$$x^2 + y^2 = \text{Distance}^2$$

4. Vertex Distance: The closest distance to the focus is the distance to the vertex:

$$a = 75,000 \cdot \frac{2 - \sqrt{3}}{2}$$

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Summary of Steps

1. Define the parabola, setting Earth at the origin and deriving the equation for a parabolic trajectory.
2. Use trigonometry to relate $x$ to the given angle and distance.
3. Substitute into the parabola’s focus-directrix definition to solve for $a$, the vertex distance.
4. Conclude that the closest distance of the comet to the Earth is approximately $10,048.095 , \mathrm{km}$.

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