Question

$\begin{array}{cccccc}& \text{Column - I}& & \text{Column - II}& & \text{Column - III}\\ \left(I\right)& y.(y^{\prime }{)}^2-x{y}^{\prime }(1+y)+x^2=& \left(i\right)& \left[y\right]=1,\text{where [.] is greatest}& \left(p\right)& \text{Curve is bounded with area},\pi \\ & 0,y\left(\sqrt{3}\right)=2& & \text{integer function}& & \\ \left(II\right)& y^{\prime }=\frac{y^2-x^2}{2xy},y\left(1\right)=1& \left(ii\right)& \text{Maximum value of y is 3}& \left(Q\right)& \text{Area bounded by curve in first quadrant with}\\ & & & & & \text{co-ordinate axes is}\frac{3\pi }{4}\\ \left(III\right)& y^{\prime }=\frac{-9x}{y},y\left(1\right)=0& \left(iii\right)& \text{Maximum value of y is not defined}& \left(R\right)& \text{Curve is conic with eccentricty},\frac{1}{2}\\ \left(IV\right)& y^{\prime }=\frac{x}{y},y\left(2\right)=0& \left(iv\right)& \text{Maximum value of y is 1}& \left(S\right)& \text{Curve is conic with eccentricity},\sqrt{2}\end{array}$

The correct combination is

• A. (I) (iii) (S)
• B. (II) (ii) (R)
• C. (III) (iv) (Q)
• D. (IV) (iv) (S)

Answer 1

The correct option is A

(I) (iii) (S)

(I) $y(y^{\prime }{)}^2=x{y}^{\prime }(1+y)+x^2=0$
$\Rightarrow (y{y}^{\prime }-x)(y^{\prime }-x)=0$
$\Rightarrow y{y}^{\prime }=x$ or $y^{\prime }=x$
$\Rightarrow y^2=x^2+C$ $y=\frac{x^2}{2}+C$
$y\left(\sqrt{3}\right)=2$
$\Rightarrow y^2=x^2+1$

Max. value of y is not defined.
Eccentricity $=\sqrt{2}$

(II) $y^{\prime }=\frac{y^2-x^2}{2xy};y\left(1\right)=1$
$\frac{dy}{dx}=\frac{1}{2}(\frac{y}{x}-\frac{x}{y})$
It is a homogeneous differential equation

$y^2+x^2=cx$
$y\left(1\right)=1\Rightarrow 1+1=c\Rightarrow c=2$
$\therefore x^2+y^2-2x=0$
Maximum value of $y$ is 1
Area $=\pi r^2=\pi$

(III)

$y^{\prime }=-\frac{9x}{y},y\left(1\right)=0$
$\Rightarrow \int ydy+\int 9xdx=\int 0$
$\Rightarrow \frac{y^2}{2}+9x^2=8\Rightarrow x^2+\frac{y^2}{9}=1$
Maximum value of $y$ is 3
Area $=3\pi$

(IV) $y^{\prime }=\frac{x}{y};y\left(2\right)=0$
$\int ydy=\int xdx$
$\Rightarrow y^2=x^2-4$
$\Rightarrow x^2-y^2=4$
Maximum value of $y$ is not defined and eccentricity $=\sqrt{2}$

Thus, the correct combination is (I) (iii) (S)