Let e 1 and e 2 be the eccentricities of the ellipse x 2 25 + y 2 b 2 = 1   b < 5 and the hyperbola x 2 16 - y 2 b 2 = 1 respectively satisfying e 1 e 2 = 1 . If α and β are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the…

Question

Let e 1 and e 2 be the eccentricities of the ellipse x 2 25 + y 2 b 2 = 1 &#160; b &#60; 5 and the hyperbola x 2 16 - y 2 b 2 = 1 respectively satisfying e 1 e 2 = 1 . If &#945; and &#946; are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair &#945; , &#160; &#946; is equal to:

Aicharya · Accepted Answer

e 1 = 1 - b 2 25 and e 2 = 1 + b 2 16 Given e 1 e 2 = 1 &#8658; e 1 e 2 2 = 1 &#8658; 1 - b 2 25 1 + b 2 16 = 1 &#8658; b 2 16 - b 2 25 - b 4 25 &#215; 16 = 0 &#8658; 9 16 &#215; 25 b 2 - b 4 25 &#215; 16 = 0 &#8658; b 2 = 9 Thus, e 1 = 1 - 9 25 = 4 5 and e 2 = 1 + 9 16 = 5 4 Therefore, &#945; = 2 5 e 1 = 8 and &#946; = 2 4 e 2 = 10 &#8756; &#945; , &#160; &#946; = 8 , &#160; 10

Let e 1 and e 2 be the eccentricities of the ellipse x 2 25 + y 2 b 2 = 1   b < 5 and the hyperbola…

Solution

Let e 1 and e 2 be the eccentricities of the ellipse x 2 25 + y 2 b 2 = 1 &#160; b &#60; 5 and the hyperbola…

Solution

Let e 1 and e 2 be the eccentricities of the ellipse x 2 25 + y 2 b 2 = 1 b < 5 and the hyperbola…