Draw the diagram for the given situation: "A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. When the comet is 80 million kms from the sun, the line segment from the sun to the comet makes an angle of $\dfrac{\pi}{3}$ radians with the axis of the orbit."

![Figure — A rightward-opening parabola with vertex V at the origin and axis along the positive x-axis; mark th](/figures/conic-sections/parabola-comet-orbit-y2-eq-80x-focus-sun-point-p-2ki7zd.svg) > [!TLDR] > Using the focus-polar equation of a parabola $r=\dfrac{l}{1-\cos\theta}$ with the given $(r,\theta)$ point gives the semi-latus rectum, hence the orbit's Cartesian equation $y^2=80x$ and a closest approach of 20 million km. > > 1. Model the orbit as a parabola with the sun $S$ at the focus. Take $S$ as pole and the axis of the parabola as the initial line, oriented so the curve opens away from the sun (comet arrives from far away and swings past the sun). > 2. The polar equation of a parabola referred to its focus is $r=\dfrac{l}{1-\cos\theta}$, where $l$ is the semi-latus rectum ($l=2a$ for $y^2=4ax$). > 3. Substitute the given data $r=80$, $\theta=\pi/3$: $\cos(\pi/3)=\tfrac12$, so $80=\dfrac{l}{1-\tfrac12}=\dfrac{l}{\tfrac12}=2l$. > 4. Hence $l=40$ (million km), so $a=\dfrac{l}{2}=20$. > 5. With the vertex $V$ taken as the Cartesian origin and the axis along the x-axis, the orbit is $y^2=4ax=80x$. …

Draw the diagram for the given situation: "A comet is moving in a parabolic orbit around the sun which is at t

Question 98 of 126

\dfrac{\pi}{3}

Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2019Subjective· 2mImportance★★★★★

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Figure — A rightward-opening parabola with vertex V at the origin and axis along the positive x-axis; mark th

Using the focus-polar equation of a parabola $r=\dfrac{l}{1-\cos\theta}$ with the given $(r,\theta)$ point gives the semi-latus rectum, hence the orbit's Cartesian equation $y^2=80x$ and a closest approach of 20 million km.

Model the orbit as a parabola with the sun $S$ at the focus. Take $S$ as pole and the axis of the parabola as the initial line, oriented so the curve opens away from the sun (comet arrives from far away and swings past the sun).
The polar equation of a parabola referred to its focus is $r=\dfrac{l}{1-\cos\theta}$ , where $l$ is the semi-latus rectum ( $l=2a$ for $y^2=4ax$ ).
Substitute the given data $r=80$ , $\theta=\pi/3$ : $\cos(\pi/3)=\tfrac12$ , so $80=\dfrac{l}{1-\tfrac12}=\dfrac{l}{\tfrac12}=2l$ .
Hence $l=40$ (million km), so $a=\dfrac{l}{2}=20$ .
With the vertex $V$ taken as the Cartesian origin and the axis along the x-axis, the orbit is $y^2=4ax=80x$ . …

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Q.Draw the diagram for the given situation: "A comet is moving in a parabolic orbit around the sun which is at the focus of a parabola. When the comet is 80 million kms from the sun, the line segment from the sun to the comet makes an angle of $\dfrac{\pi}{3}$ radians with the axis of the orbit."

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