Statement $$1$$: If the extremities of the latus rectum of the ellipse $$x2a2+y2b2=1$$ (a$$ $$b), having positive ordinates lies on the parabola $$x2=$$−$$2(y$$−$$2)$$, then, $$a=2Statement$$ $$2$$: If the length of the latus rectum of the ellipse $$x2a2+y2b2=1$$ is equal to the distance between the foci, then the eccentricity e of the ellipse satisfies $$e2+e$$−$$1=0$$

Question

Statement $$1$$: If the extremities of the latus rectum of the ellipse   $$x2a2+y2b2=1$$ (a$$> $$b), having positive ordinates lies on the parabola $$x2=$$−$$2(y$$−$$2)$$,  then,  $$a=2Statement$$ $$2$$: If the length of the latus rectum of the  ellipse  $$x2a2+y2b2=1$$  is equal to the distance between the foci, then the eccentricity e of the ellipse satisfies $$e2+e$$−$$1=0$$

Infinity Learn · Accepted Answer

$$Statement-2$$ is true by definition of conjugate diameters.  Let $$y=mx$$ and $$y=m$$′x x be two conjugate diameters ofthe ellipse $$x2a2+y2b2=1$$.  Let  (h$$,$$k) be the mid point of chord whose step is m then  $$hxa2+kyb2$$−$$1=h2a2+k2b2$$−$$1$$⇒$$m=$$−$$b2a2hk$$⇒ Locus of  (h$$, $$k)  is $$y=$$−$$b2a2mx$$⇒−$$b2a2m=m$$′⇒m′$$m=$$−$$b2a2$$  . $$(using$$ $$statement-2)and$$ thus $$statement-$$ I is false.

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