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The line y=m x-((a^2-b^2)m)/(sqrt(a^2+b^...

The line `y=m x-((a^2-b^2)m)/(sqrt(a^2+b^2m^2))` is normal to the ellise `(x^2)/(a^2)+(y^2)/(b^2)=1` for all values of `m` belonging to `(0,1)` (b) `(0,oo)` (c) `R` (d) none of these

A

(0,1)

B

`(0,oo)`

C

R

D

none of these

Text Solution

Verified by Experts

The eqquation of the normal to the given ellipse at the point `P(a cos theta, b sin theta) "is" ax sec theta- "by cosec" theta=a^(2)-b^(2)`. Then,
`y=((a)/(b) tan theta)x-((a^(2)-b^(2)))/(b) sin theta" "(1)`
Let `(a)/(b) tan theta=m`
so that
`y=((a)/(b) tan theta)x-((a^(2)-b^(2)))/(b) sin theta" "(1)`
Hence, the equation of the normal equation (1) becomes
`y=mx-((a^(2)-b^(2)))/(sqrt(a^(2)+b^(2)m^(2)))` ltbrlt `:. m in R, "as" m=(a)/(b) tan theta in R`
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