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If B and C Are Lengths of the Segments of Any Focal Chord of the Parabola Y2 = 4ax, Then Write the Length of Its Latus-rectum. - Mathematics

If b and c are lengths of the segments of any focal chord of the parabola y2 = 4ax, then write the length of its latus-rectum. 

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Let S (a, 0) be the focus of the given parabola.
Let the end points of the focal chord be  \[P \left( a t^2 , 2at \right) and Q\left( \frac{a}{t^2}, \frac{- 2a}{t} \right)\] 

SP and SQ are segments of the focal chord with lengths b and c, respectively.
∴ SP = b, SQ = c 

\[Also, SP = \sqrt{\left( a - a t^2 \right) + 4 a^2 t^2} = a\left( 1 + t^2 \right)\]
\[And, SQ = \sqrt{\left( a - \frac{a}{t^2} \right) + \frac{4 a^2}{t^2}} = a\left( 1 + \frac{1}{t^2} \right)\]

Now, we have: 

\[\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a\left( 1 + t^2 \right)} + \frac{t^2}{a\left( 1 + t^2 \right)} = \frac{1}{a}\] 

\[\Rightarrow \frac{1}{b} + \frac{1}{c} = \frac{1}{a}\]
\[ \Rightarrow \frac{b + c}{bc} = \frac{1}{a}\]
\[ \Rightarrow a = \frac{bc}{b + c}\] 

∴ Length of the latus rectum = 4a = \[\frac{4bc}{b + c}\]


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RD Sharma Class 11 Mathematics Textbook
Chapter 25 Parabola
Exercise 25.2 | Q 6 | Page 28
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