1.

thetangent andnormaltothetangentto theellipse(x^(2))/(a^(2))+(y^(2))/(b^(2))=1at thepoint(3,(-9)/(2)),thenthe area( in sq units ) oftheDeltaPQRIs

Answer»

`(16)/(3)`
`(14)/(3)`
`(34)/(15)`
`(68)/(15)`

Solution :Equationof givenellipse is
` 3x^(2) +5y^(2)=32`
bnow,the slopeof tangent andnormalat POINT `P(2,2)` totheellipse(i) areresectively
`m_(T)=(dy)/(dx)|_(""(2,2))and m_(N)=-(dx)/(dy)|_((2,2)"")`
ondiferntiatingellipse(i) w.R.tx, weget
`6x+10y(dy)/(dx)=0implies (dy)/(dx)=-(3x)/(5y)`
`SoM_(T) =-(3x)/(5y)|_((2,2)"")=-(3)/(5) andm_(N) = (5y)/(3Y)|_((2,2)"")=(5)/(3)`
Nowequationof tangentand normal tothegivenellipse(i) atpointP(2,2) are
`(y-2)=-(3)/(5)(x-2)`
and `(y-2)=(5)/(3)(x-2)` RESPECTIVLY.
It isgiventhat pointif INTERSECTIONOF tangentandnormalare Qand RAt X- axisRespectively.
so, `Q((16)/(3),0) and R((4)/(5) ,0)`
`therefore ` Area of`DeltaPQR =(1)/(2)(QR)xx"height "`
`=(1)/(2) xx(68)/(15)xx2=(68)/(15) sq units `
`[ :'QR= sqrt(((16)/(3)-(4)/(5))^(2))=sqrt(((68)/(15))^(2))` = (68)/(15)` andheight= 2]


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