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Vector Algebra

Question
CBSEENMA12036042
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The area (in sq units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipsestraight x squared over 9 plus straight y squared over 5 space equals space 1 space is space

  • 27/4

  • 18

  • 27/2

  • 27

Solution

D.

27

given equation of ellipse is 
straight x squared over 9 space plus space straight y squared over 5 space equals space 1 straight a squared space equals space 9 comma space straight b squared space equals 5 straight a space equals 3 comma space straight b space equals space square root of 5 straight e space equals square root of 1 minus straight b squared over straight a squared end root space equals space square root of 1 minus 5 over 9 end root space equals space 2 over 3 foci space left parenthesis space plus-or-minus space ae comma space 0 right parenthesis space space equals left parenthesis plus-or-minus 2 comma 0 right parenthesis and space straight b squared over straight a space equals space 5 over 3
Therefore, Extremities of one of latus rectum are
open parentheses 2 comma 5 over 3 close parentheses space and space open parentheses 2 comma fraction numerator negative 5 over denominator 3 end fraction close parentheses space Therefore comma space Equation space of space tangent space at space open parentheses 2 comma 5 over 3 close parentheses space is comma fraction numerator straight x left parenthesis 2 right parenthesis over denominator 9 end fraction space plus fraction numerator straight y left parenthesis 5 divided by 3 right parenthesis over denominator 5 end fraction space equals space 1 or space 2 straight x space plus space 3 straight y space equals space 9 Eq. space left parenthesis ii right parenthesis space intersects space straight X space and space straight Y space minus axes space at space open parentheses 0 comma 9 over 2 close parentheses space and space left parenthesis 0 comma 3 right parenthesis
therefore, area of quadrilateral = 4 x Area of ΔPOQ
 = 4 space straight x space open parentheses 1 half straight x 9 over 2 straight x 3 close parentheses
= 27 sq units

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