If Ellipse E_n is drawn such that it touches Ellipse E_{n-1} at the extremities of the major axis, and has it's foci at the extremities of the minor axis of E_{n-1} then answer the following Qsns....
1) Eccentricity e_n of Ellipse E_n is independent of n then find eccentricity of ellipse E_{n-3}.
2) In last case, let the axis of E_n be along Y-axis (Which axis are they referring to ?) - then find locus of mid point of chord of slope -1 of E_n.
3) Eccentricity of Ellipse E_n is e_n, then find locus of \left(e_n^2,e_{n-1}^2 \right).
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1 Answers
66Let us denote the semi-major axis of En as an, the semi-minor axis as bn and the eccentricity as en.
According to the given conditions,
b_n=a_{n-1} and a_ne_n=b_{n-1}
Now,
b_n^2=a_n^2(1-e_n^2)
from where we obtain a recurrence relation
e_n^2=\dfrac{1-e_{n-1}^2}{2-e_{n-1}^2}
From this relation the third part easily follows. The locus of (en2, en-12) is
x=\dfrac{1-y}{2-y}\quad \Rightarrow \ y = \dfrac{2x-1}{x-1}
For the first part, if en is independent of n, it must be a fixed point of the sequence {en}. If that is denoted as e, then
e^2=\dfrac{1-e^2}{2-e^2}
Solving for e, we get
e=\dfrac{\sqrt{5}-1}{2}
Hence,
e_n=e_{n-3}=\dfrac{\sqrt{5}-1}{2}
Now can you do the 2nd one..we require a diameter in this case.
