1.

निम्न वक्रों पर स्पर्शी के समीकरण ज्ञात कीजिये - (i) `x=cos theta,y=sin theta, theta=(pi)/(4)` पर (ii) `xy=4" बिन्दु "(1,4)" पर "`

Answer» (i) दिया है : `x=cos theta, y=sin theta`
`theta=(pi)/(4)," पर "x=cos(pi)/(4)=(1)/(sqrt(2))`
तथा `y=sin""(pi)/(4)=(1)/(sqrt(2))`
`:." "(x,y)=((1)/(sqrt(2)),(1)/(sqrt(2)))`
अब `" "(dx)/(d theta)=-sin theta" तथा "(dy)/(d theta)=cos theta`
`implies" "(dy)/(dx)=(dy//d theta)/(dx//d theta)=(cos theta)/(-sin theta)=-cot theta`
`:." "((dy)/(dx))_(theta=(pi)/(4)("पर"))=-cot""(pi)/(4)=-1`
`implies ` स्पर्शी का समीकरण
`y-(1)/(sqrt(2))=-1(x-(1)/(sqrt(2)))`
`implies" "y-(1)/(sqrt(2))=-x+(1)/(sqrt(2))`
`implies" "y+x=(2)/(sqrt(2))`
`implies" "x+y=sqrt(2)`
(ii) दिया है : `xy=4`
`implies" "x(dy)/(dx)+y=0`
`implies" "(dy)/(dx)=-(y)/(x)`
`:." "((dy)/(dx))_((1","4)" पर")=-(4)/(1)=-4`
`:. ` स्पर्शी का समीकरण
`y-4=-4(x-1)`
`implies" "y+4x=8`


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