1.

If `p_(1) and p_(2)` are the lengths of the perpendicular form the orgin to the line `x sec theta+y cosec theta=a and xcostheta-y sin theta=a cos 2 theta` respectively then prove that `4p_(1)^(2)+p_(2)^(2)=a^(2)`

Answer» We have `x sectheta+y "cosec" theta =a Rightarrow (x)/(cos theta)+(y)/(sin theta)-a=0.......(i)`
Now, `p_(1)` is the length of perpendicular from the origin to line (i), so we have
`p_(1)=(|(1)/(cos theta)xx0+(1)/(sin theta)xx0-a|)/(sqrt((1)/(cos^(2)theta)+(1)/(sin^(2)theta)))=|a|sin theta cos theta=(|a|)/(2)sin 2theta`
`Rightarrow 2p_(1)=|a|sin 2theta`
`Rightarrow 4p_(1)^(2)=a^(2)sin^(2) 2theta........(ii)`
The other line is `x cos theta-y sin theta-a cos 2theta=0`.......(ii)
Now, `p_(2)` is the length of perpendicular from the origin to line (ii), so we have
`P_(2)=(|0xx cos theata-0xxsin theta-acos2 theta|)/(sqrt(cos^(2)theta+sin^(2)theta))=|a cos 2theta|`
`Rightarrow p_(2)^(2)=a^(2) cos^(2)2 theta...(iv)`
Adding (ii) and (iv), we get
`4p_(1)^(2)+p_(2)^(2)=a^(2)(sin^(2)2theta+cos^(2)2theta)`
`Rightarrow 4p_(1)^(2)+p_(2)^(2)=a^(2)`


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