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Question Number 9199 by tawakalitu last updated on 22/Nov/16

If r^{2} = {(x + ea)}^{2} + y^{2} and s^{2} = {(x - ea)}^{2} + y^{2}

and r + s = 2 a, Prove that :

r = a + ex, s = a - ex, and that,

x^{2} (1 - e^{2}) + y^{2} = a^{2} (1 - e^{2})

Commented by 123456 last updated on 22/Nov/16

r^{2} + s^{2} = 2 y^{2} + {(x + e a)}^{2} + {(x - e a)}^{2}

r^{2} + s^{2} = 2 y^{2} + x^{2} + 2 e a + {(e a)}^{2} + x^{2} - 2 e a + {(e a)}^{2}

r^{2} + s^{2} = 2 [x^{2} + y^{2} + {(e a)}^{2}]

r + s = 2 a

{(r + s)}^{2} = 4 a^{2}

r^{2} + s^{2} + 2 r s = 4 a^{2}

2 [x^{2} + y^{2} + {(e a)}^{2} + r s] = 4 a^{2}

x^{2} + y^{2} + {(e a)}^{2} + r s = 2 a^{2}

Commented by tawakalitu last updated on 23/Nov/16

I appreciate sir . God bless you .