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Question Number 51614 by peter frank last updated on 29/Dec/18

P r o v e t h a t l e n g t h o f

l a c t u s r e c t u m w h e n

d i r e c t r i x a r e p a r a l l e l t o

y - a x i s i s

y_{1} - y_{2} = \frac{2 b^{2}}{a}

Answered by tanmay.chaudhury50@gmail.com last updated on 29/Dec/18

f o r e l l i p s e \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 d i t e c t r i x i s ∥ t o y a x i s

f o c u s (c, 0) a n d (- c, 0) s o e q n o f s t l i n e ∥ t o y a x i s

p a s s i n g t h r o u g h (c, 0) i s x = c

s o l v e x = c a n d \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

\frac{y^{2}}{b^{2}} = 1 - \frac{c^{2}}{a^{2}} s o y = \pm \frac{b}{a} \sqrt{a^{2} - c^{2}}

n o w d i z t a n c e b e t w e e n (c, \frac{b}{a} \sqrt{a^{2} - c^{2}}) a n d (c, \frac{- b}{a} \sqrt{a^{2} - c^{2}})

i s = \frac{2 b}{a} \sqrt{a^{2} - c^{2}} [f o r e l l i p s e c^{2} = a^{2} - b^{2}]

= \frac{2 b}{a} \times b = \frac{2 b^{2}}{a}

f o r h y p e r b o l a \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

d i r e c t r i x i s ∥ t o y a x i s

f o c u s (c, 0) (- c, 0)

e q n o f s t l i n e p a s s i n g t h r o u g h (c, 0) a n d ∥ t o y a x i s

i s x = c

s o l v e x = c a n d \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1

\frac{y^{2}}{b^{2}} = \frac{c^{2}}{a^{2}} - 1

y = \pm \frac{b}{a} \sqrt{c^{2} - a^{2}}

y = \pm \frac{b}{a} \times b [f o r h y p e r b o l a c^{2} = a^{2} + b^{2}]

y = \pm \frac{b^{2}}{a}

s o l a t u s r e c t u m i s \frac{2 b^{2}}{a}

Commented by peter frank last updated on 29/Dec/18

v e r y n i c e s i r \dots . t h a n k s

Answered by peter frank last updated on 29/Dec/18

\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1

x = h + a e (p a r a l l e l t o y - a x i s)

\frac{{(h + a e - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1

e^{2} - 1 = \frac{{(y - k)}^{2}}{b^{2}}

y - k = \sqrt{b^{2} (e^{2} - 1)}

y - k = \pm b \sqrt{(e^{2} - 1)}

y = k \pm b \sqrt{(e^{2} - 1)}

y_{1} = k - b \sqrt{e^{2} - 1} \dots . (i)

y_{2} = k + b \sqrt{e^{2} - 1} \dots . (i i)

y_{2} - y_{1} = - 2 b \sqrt{e^{2} - 1}

y_{1} - y_{2} = 2 b \sqrt{e^{2} - 1}

f r o m

b^{2} = a^{2} (e^{2} - 1)

y_{1} - y_{2} = 2 b \times \frac{b}{a}

y_{1} - y_{2} = \frac{2 b^{2}}{a}