Mathematics

Mathematics

Problem

Let

E_1

and

E_2

be two ellipses whose centers are at the origin. The major axes of

E_1

and

E_2

lie along the

X

-axis and the

Y

-axis, respectively. Let

S

be the circle

x^2 + (y-1)^2 = 2

. The straight line

x + y = 3

touches the curves,

S

,

E_1

and

E_2

at

P

,

Q

and

R

respectively. Suppose that

PQ = PR = \frac{2\sqrt{2}}{3}

. If

e_1

and

e_2

are the eccentricities of

E_1

and

E_2

, respectively, then the correct expression(s) is (are) a.

e_1^2 + e_2^2 = \frac{43}{40}

b.

e_1 e_2 = \frac{\sqrt{7}}{2\sqrt{10}}

c.

|e_1^2 - e_2^2| = \frac{5}{8}

d.

e_1 e_2 = \frac{\sqrt{3}}{4}

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