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In a parabola, Prove that the length of a focal chord which is inclined at ${30^0}$ to the axis is four times the length of the latus-rectum.

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Last updated date: 16th May 2024
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Answer
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Hint: - Use equation of parabola in polar form, \[\dfrac{{2a}}{r} = 1 - \cos \theta \]

Equation of parabola in polar form is
\[\dfrac{{2a}}{r} = 1 - \cos \theta ..........................\left( b \right)\]
Where $r$ is the distance between focus and parametric point.
As we know latus rectum of parabola is \[{\text{ = 4a}}\]
Let PP’ be the focal chord and it is given that it is inclined at ${30^0}$ then parametric angles of P and P’ are ${30^0}$and $\pi + {30^0}$ respectively.
Let S be the focus which divide the focal chord into two equal parts
I.e. \[{\text{PS + SP' = PP'}}\]…………………(c)
$ \Rightarrow r = PS = SP'$
From equation (b)
\[\begin{gathered}
   \Rightarrow \dfrac{{2a}}{{PS}} = 1 - \cos {30^0} \\
   \Rightarrow PS = \dfrac{{2a}}{{1 - \cos {{30}^0}}}....................\left( 1 \right) \\
\end{gathered} \]
From equation (b)
\[\begin{gathered}
   \Rightarrow \dfrac{{2a}}{{SP'}} = 1 - \cos \left( {\pi + {{30}^0}} \right) = 1 + \cos {30^0} \\
   \Rightarrow SP' = \dfrac{{2a}}{{1 + \cos {{30}^0}}}................\left( 2 \right) \\
\end{gathered} \]
Now add equation (1) and (2) and from equation (c)
\[\begin{gathered}
   \Rightarrow {\text{PS + SP' = PP' = }}\dfrac{{2a}}{{1 - \cos {{30}^0}}} + \dfrac{{2a}}{{1 + \cos {{30}^0}}} \\
   \Rightarrow {\text{PP' = }}\dfrac{{4a}}{{1 - {{\cos }^2}{{30}^0}}} = \dfrac{{4a}}{{1 - \dfrac{3}{4}}} = 16a \\
\end{gathered} \]

So, length of focal chord \[{\text{PP}}' = 16a = 4 \times 4a{\text{ = 4}} \times {\text{Latus - Rectum}}\]
Hence Proved.

Note: - In such types of questions the key concept we have to remember is that always remember the equation of parabola in polar form and its parametric angles which is stated above and also remember the value of latus rectum of the parabola, then use equation (c) to get the required length of the focal chord which is four times the latus rectum.