Question

Pramole # 10 : Find the equation of axes, directrix, co-ordinate of focii, centre, vertices, length of latus - rectum and eccentricity of an ellipse 16x2 + 25y2 - 96x - 100 y + 156 = 0.

Railway Exams

Maths

Solution

151

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The given ellipse is ( frac{(x-3)^{2}}{25}+frac{(y-2)^{2}}{16}=1 ) Let ( x-3=x, y-2=Y, ) so equation of ellipse becomes as ( frac{x^{2}}{5^{2}}+frac{y^{2}}{4^{2}} ) equation of major axis is ( Y=0 ) ( Rightarrow quad y=2 ) equation of minor axis is ( X=0 quad Rightarrow=3 ). centre ( (X=0, Y=0) ) ( Rightarrow quad x=3, y=2 ) [ C equiv(3,2) ] Length of semi-major axis a ( =5 ) Length of major axis ( 2 a=10 ) Length of semi-minor axis ( b=4 ) Length of minor axis ( =2 b=8 ) Let 'e' be eccentricity ( therefore ) ( b^{2}=a^{2}left(1-e^{2}right) ) ( e=sqrt{frac{a^{2}-b^{2}}{a^{2}}}=sqrt{frac{25-16}{25}}=frac{3}{5} ) Length of latus rectum ( =L L^{prime}=frac{2 b^{2}}{a}=frac{2 times 16}{5}=frac{32}{5} ) Co-ordinates focii are ( X=pm a e, Y=0 ) ( Rightarrow quad S equiv(X=3, Y=0) ) ( &^{prime} equiv(X=-3, Y=0) ) ( S equiv(6,2) ) ( &^{prime} equiv(0,2) )