- Home
- Standard 11
- Mathematics
यदि $x^{2}+9 y ^{2}-4 x+3=0, x, y \in R$ हैं, तो $x$ तथा $y$ क्रमशः निम्न में से किस अंतराल में है?
$\left[-\frac{1}{3}, \frac{1}{3}\right]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$
$\left[-\frac{1}{3}, \frac{1}{3}\right]$ and $[1,3]$
$[1,3]$ and $[1,3]$
$[1,3]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$
Solution
$x^{2}+9 y^{2}-4 x+3=0$
$\left(x^{2}-4 x\right)+\left(9 y^{2}\right)+3=0$
$\left(x^{2}-4 x+4\right)+\left(9 y^{2}\right)+3-4=0$
$(x-2)^{2}+(3 y)^{2}=1$
$\frac{(x-2)^{2}}{(1)^{2}}+\frac{y^{2}}{\left(\frac{1}{3}\right)^{2}}=1$ (equation of an ellipse).
As it is equation of an ellipse, $x \,\&\, y$ can vary inside the ellipse.
So, $x-2 \in[-1,1]$ and $y \in\left[-\frac{1}{3}, \frac{1}{3}\right]$
$x \in[1,3] y \in\left[-\frac{1}{3}, \frac{1}{3}\right]$
