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Question

# Let E1 and E2 be two ellipse whose centres are at the origin. The major axes of E1 and E2 lie along x-axis and y-axis respectively. Let S be the circle x2+(y−1)2=2 the straight line x+y=3 touches the curves S, E1 and E2 at P, q and R, respectively. Suppose that PQ=PR=2√23. If e1 and e2 are the eccentricities of E1 and E2 respectively, then the correct expression(s) is/are

A

e12+e22=4340

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B

e1e2=7210

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C

|e12e22|=58

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D

e1e2=34

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Solution

## The correct options are A e12+e22=4340 B e1e2=√72√10 Here E1:x2a2+y2b2=1,(a>b)E2:x2c2+y2d2=1,(c<d) and S:x2+(y−1)2=2 As tangent to E1, E2 and S is x+y=3 Let the point of contact of tangent be (x1,y1) to S.∴x.x1+y.y1−(y+y1)+1=2Orxx1+yy1−y=(1+y1),is same asx+y=3⇒x11=y1−11=1+y13i.e.x1=1 and y1=2p=(1,2)PR=PQ=2√23thus by parametric form,x−1−1√2=y−21√2=±2√23⇒(x=53,y=43)And(x=13,y=83) ∴Q=(53,43) and R=(13,83) Now, equation tangent at Q on Ellipse E1 is x.5a2.3+y.4b2.3=1 On comparing with x+y=3, we get a2=5 and b2=4 ∴e21=1−b2a2=1−45=15 Also, equation of tangent at R on ellipse E2 is x.1a2.3+y.8b2.3=1 On comparing with x+y=3, we get a2=1,b2=8 ∴e22=1−a2b2=1−18=78 Now, e21.e22=740⇒e1e2=√72√10 and e21+e22=15+78=4340 Also, ∣∣e21−e22∣∣=∣∣15−78∣∣=2740

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