Question

Let R =$\left\{\right(x,y):x,y\in R,x^2+y^2\le 25\}{R}^{\prime }=\left\{\right(x,y):x,y\in R,y\ge \frac{4}{9}{x}^2\}$then

• A. $DomainR\cap R^{\prime }=[-3,3]$
• B. $RangeR\cap R^{\prime }\supset [0,4]$
• C. $RangeR\cap R^{\prime }=[0,5]$
• D. $definesafunctionR\cap R^{\prime }$

Answer 1

Answer: (A), (C)

The correct options are
A

$DomainR\cap R^{\prime }=[-3,3]$

C

$RangeR\cap R^{\prime }=[0,5]$

The equation $x^2+y^2=25$ represents a circle with centre (0,0) and radius 5 and the equation y =$\frac{4}{9}{x}^2$ represents a parabola with vertex (0,0) and focus $(0,\frac{9}{16})$. Hence $R\cap R^{\prime }$ is the set of point indicated in the fig.

= $\left\{\right(x,y):-3\le x\le ,0\le y\le 5\}$

Thus dom $R\cap R^{\prime }=[-3,3]$ and range $R\cap R^{\prime }=[0,5]\supset [0,4]$

Since (0,0) $\in$ R $\cap R^{\prime }$ and (0, 0) $\in$ R $\cap R^{\prime }$

Hence $\in$ R$\cap R^{\prime }$ doesn't define a function.