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Q.

Suppose that the foci of the ellipse x29+y25=1 are f1,0 and f2,0 where f1>0 and f2<0.          Let P1 and P2 be two parabolas with a common vertex at 0,0 and with foci at  f1,0 and 2f2,0, respectively. Let T1 be a tangent to P1 which passes through 2f2,0  and T2 be a tangent to P2  which passes through f1,0. If  m1is the slope of T1 and m2  is the slope of T2 then the value of 1m12+m22 is

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answer is 4.

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Detailed Solution

Wehave e2=1-b2a2=1-59=49 ⇒e=23∴f1,0=2,0  and f2,0=−2,0⇒P1⇒y2=8x and     P2⇒y2=−16x∴Equations of tangents T1 and T2 satisfy the following conditioins0=−4m1+2m1,   2m2−4m2=0⇒m12=12,  m22=2⇒1m12=2,   1m22=4'⇒1m12+m22=4
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