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Find the equations and length of major and minor axes of $5x^{2}+9y^{2}+10x-36y-4=0$

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Toolbox:
  • Shifting of the origin from $(0,0)$ to $(h,k)$ then if the new axes form $X-Y$ coordinate system then $x=X+a$,$y=Y+k$
  • The new coordinates of a point $P(x,y)$ become $X=x-a,Y=y-k$
  • Standard forms of equation of the ellipse with major axes 2a, minor axis 2b $(a >b)$ eccentricity e and $b^2=a^2(1-e^2)$ or $e^2=1-\large\frac{b^2}{a^2}$
  • $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$
  • http://clay6.com/mpaimg/6_2_Toolbox.png
  • Foci$(\pm ae,o),$ center $(0,0)$,vertices $(\pm a,0)$
  • End points of Latus Rectum $\; (ae,\pm \large\frac{b^2}{a})$ and $(-ae,\pm \large\frac{b^2}{a})$
  • Directrices $x=\pm \large\frac{a}{e}$.
  • The major axis is $y=0$ (x- axis) and the minor axes is $x=0$ (y- axis)
  • $\large\frac{x^2}{b^2}+\frac{y^2}{a^2}$$=1$
  • http://clay6.com/mpaimg/6_2_Toolbox1.png
  • Foci$(0,\pm ae),$ center $(0,0)$,vertices $(0,\pm a)$
  • End points of Latus Rectum $(\pm \large\frac{b^2}{a}$$,ae)$ and $(\pm \large\frac{b^2}{a}$$,-ae)$
  • Directrices $y=\pm \large\frac{a}{e}$.
  • The major axis is $x=0$ (y- axis) and the minor axes is $y=0$ (x- axis)
  • General form of standard ellipses with centre $C$ with major axis $2a$,minor axis $2b$ and axes parallel to the coordinate axes.
  • $\large\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}$$=1$
  • In this case major axis y=k and minor axis $x=h$.
  • $\large\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}$$=1$
  • In this case major axis $x=h$ and minor axis $y=k$.
Step 1:
$5x^2+10x+9y^2-36y=4$
Completing squares,
$5(x^2+2x+1)+9(y^2-4y+4)=4+5+36$
$5(x+`)^2+9(y-2)^2=45$
Step 2:
Dividing the above equation by $45$ we get
$\large\frac{(x+1)^2}{9}+\frac{(y-2)^2}{5}$$=1$
$a^2=9,b^2=5$
$\Rightarrow a=3,b=\sqrt 5$
Step 3:
Length of major axis = $2a=2\times 3=6$
Length of minor axis = $2b=2\times \sqrt 5=2\sqrt 5$
Equation of major axis =$y = 2$
Equation of minor axis =$x = -1$
answered Jun 17, 2013 by sreemathi.v
 

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