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Which of the following pair(s) of family is/are orthogonl? where c and k are arbitrary constant.A. `16x^(2)+y^(2)=c and y^(16)=kx`B. `y=x+ce^(-x) and x+2=y+ke^(-y)`C. `y=cx^(2) and x^(2)+2y^(2)=k`D. `x^(2)-y^(2)=c and xy =k` |
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Answer» Correct Answer - A::B::C::D (a) `16x^(2)+y^(2)=c rArr(dy)/(dx)=m_(1)=-(16x)/(y)` `y^(16)=kx" "rArr " "(dy)/(dx)=m_(2)=(k)/(16y^(15))` `m_(1)m_(2)=-(16x)/(y).(k)/(16y^(15))` `=-(x)/(y^(16)).k` `=-(x)/(y^(16)).(y^(16))/(x)=-1` `rArr" Curves are orthogonal."` (b) `y=x+ce^(-x)` `rArr" "(dy)/(dx)=1-ce^(-x)=1-(y-x)=-(y-x-1)` `x+2=y+ke^(-y)` `rArr" "(dy)/(dx)-k(dy)/(dx)e^(-y)=1` `rArr" "(dy)/(dx)[1-ke^(-y)]=1` `"or "[1-(x+2-y)](dy)/(dx)=1` `rArr" "(dy)/(dx)=m_(2)=(1)/(y-x-1)` `rArr: "m_(1)m_(2)=-1` `rArr" Curves are orthogonal."` (c) `y=cx^(2)` `rArr" "(dy)/(dx)=2cx=2x(y)/(x^(2))=(2y)/(x)=m_(1)` `"and "x^(2)+2y^(2)=k` `rArr" "2x+4y(dy)/(dx)=0` `rArr" "(dy)/(dx)=-(x)/(2y)=m_(2)` `rArr" "m_(1)m_(2)=-1` `rArr" Curves are orthogonal."` (c) `x^(2)-y^(2)=c rArr (dy)/(dx)=(x)/(y)=m_(1)` `xy=k rArr (dy)/(dx)=-(y)/(x)=m_(2)` `therefore" "m_(1)m_(2)=-1` `rArr" Curves are orthogonal."` |
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