यदि `2ycostheta=xsinthetaand 2xsectheta-y"co s ec"theta=3` और हो तो `x^(2)+4y^(2)` का मान होगा ?

Options

A. 1
B. 3
C. 2
D. 4

Answer

d

Explanation

` 2ycostheta=x sin theta`.........(i)
or `4y^(2)cos^(2)theta=x^(2)sin^(2)theta` .......(ii)
and
`2xsectheta-yco s ec theta=3sin theta cos theta`
from eq. (i) put `ycos theta=(xsin theta)/(2)`
`2xsin theta-(xsin theta)/(2)=3sin thetacos theta`
`4x sin theta-x sin theta=6 sin thetacos theta`
`x=2cos theta` ........(iii)
Now,
`x^(2)+4y^(2)=(2costheta)^(2)+(x^(2)sin^(2)theta)/(cos^(2)theta)`
from eq (ii)
(from eq (iii) `x=2costheta`)
`=4cos^(2)theta+(4cos^(2)thetasin^(2)theta)/(cos ^(2)theta)`
`=4(cos^(2)theta+sin^(2)theta)`
=4
Alternate
`2ycostheta=2sintheta`
Put `theta=45^(@)`
`2y.(1)/sqrt(2)=x(1)/sqrt(2)`
`2y=x`.......(i)
`2xsec theta-yco s ec theta=3`
but `-theta=45`
`2x.sqrt(2)-ysqrt(2)=3`
`2pi-y=(3)/sqrt(2)`.......(ii)
from eq. (i) and (ii)
` 2x-4y=0`
`(2x -y=(3)/(2))/(3y=(3)/sqrt(2))`
`y=(1)/sqrt(2)`
put value of y is eq(i)
`x=2xx(1)/sqrt(2)=sqrt(2)`
Now,
`x^(2)+4y^(2)=(sqrt(2))^(2)+4((1)/sqrt(2))^(2)`
=2+2=4