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prove that the equation of eccentricity of an ellipse can be obtained by equation e^4+e^2-1=0 if the normal drawn from one end of latus rectum of an ellipse passes through other end of the minor axis

prove that the equation of eccentricity of an ellipse can be obtained by equation e^4+e^2-1=0 if the normal drawn from one end of latus rectum of an ellipse passes through other end of the minor axis

Grade:12th pass

1 Answers

Arka Bhattacharya
11 Points
7 years ago
Let the equation of ellipse be x^2/a^2 + y^2/b^2 =1Slope of normal to ellipse at (x1,y1)= (a^2*y1)/(b^2*x1)Slope of normal at L(ae,b^2/a)=1/e (using above equation)equation of normal to ellipse at L is (ax/e) -ay=a^2 - b^2 (using y-y1=m(x-x1))Since it passes through B(0,-b), substituting in above equation of normal, we getab =a^2 - b^2And b= a*sqrt(1-e^2)finally we get sqrt(1-e^2) = e^2Squaring both sides, we get e^4 + e^2 -1= 0

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