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10.2 parabola,ellipse,hyperbola — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS10.2 parabola,ellipse,hyperbola1 Mark
MCQ
If $x^{2}+9 y^{2}-4 x+3=0, x, y \in R$, then $x$ and $y$ respectively lie in the intervals:
  • A
    $\left[-\frac{1}{3}, \frac{1}{3}\right]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$
  • B
    $\left[-\frac{1}{3}, \frac{1}{3}\right]$ and $[1,3]$
  • C
    $[1,3]$ and $[1,3]$
  • ✓
    $[1,3]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$
✓

Answer

Correct option: D.
$[1,3]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$
d
 

$x^{2}+9 y^{2}-4 x+3=0$

$\left(x^{2}-4 x\right)+\left(9 y^{2}\right)+3=0$

$\left(x^{2}-4 x+4\right)+\left(9 y^{2}\right)+3-4=0$

$(x-2)^{2}+(3 y)^{2}=1$

$\frac{(x-2)^{2}}{(1)^{2}}+\frac{y^{2}}{\left(\frac{1}{3}\right)^{2}}=1$ (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 $y \in\left[-\frac{1}{3}, \frac{1}{3}\right]$

$x \in[1,3] y \in\left[-\frac{1}{3}, \frac{1}{3}\right]$

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