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Let E1a n dE2, respectively, be two elli...

Let `E_1a n dE_2,` respectively, be two ellipses `(x^2)/(a^2)+y^2=1,a n dx^2+(y^2)/(a^2)=1` (where `a` is a parameter). Then the locus of the points of intersection of the ellipses `E_1a n dE_2` is a set of curves comprising two straight lines (b) one straight line one circle (d) one parabola

A

two straigths

B

one straiths line

C

one circle

D

one parabola

Text Solution

Verified by Experts

Let P(h,k) be the point of intersection of `E_(1) and E_(2)`. Then ,
`(h^(2))/(a^(2))+k^(2)=1`
or `h^(2)=a^(2)(1-k^(2)) " "(1)`
and `(h^(2))/(1)+(k^(2))/(a^(2))=1`
`or k^(2)=a^(2)(1-h^(2))" "(2)`
Eliminating from (1) and (2), we get
`(h^(2))/(1-k^(2))=(k^(2))/(1-h^(2))`
or `h^(2)(1-h^(2)=k^(2)(1-k^(2))`
or `(h-k)(h+k)(h^(2)+k^(2)-1)=0`
Hence, hte locus os a set of curves consisting of the straight lines y=x,y=-x, and the cirlce`x^(2)+y^(2)=1`
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