For the ellipse 4(x-2y+1)^2 + 9(2x+y+2)^2 = 2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The given equation is $4(x-2y+1)^2 + 9(2x+y+2)^2 = 25$. Divide by $25$: $\frac{(x-2y+1)^2}{25/4} + \frac{(2x+y+2)^2}{25/9} = 1$. Let $X = \frac{x-

Vedclass · Accepted Answer

equation of major axis is $x-2y+1=0$

Vedclass · Answer

(A) The given equation is $4(x-2y+1)^2 + 9(2x+y+2)^2 = 25$. Divide by $25$: $\frac{(x-2y+1)^2}{25/4} + \frac{(2x+y+2)^2}{25/9} = 1$. Let $X = \frac{x-2y+1}{\sqrt{1^2+(-2)^2}} = \frac{x-2y+1}{\sqrt{5}}$ and $Y = \frac{2x+y+2}{\sqrt{2^2+1^2}} = \frac{2x+y+2}{\sqrt{5}}$. The equation becomes $\frac{5X^2}{25/4} + \frac{5Y^2}{25/9} = 1$,which simplifies to $\frac{X^2}{5/4} + \frac{Y^2}{5/9} = 1$. Here $a^2 = 5/4$ and $b^2 = 5/9$,so $a = \sqrt{5}/2$ and $b = \sqrt{5}/3$. The major axis corresponds to the term with the larger denominator,which is $X=0$,i.e.,$x-2y+1=0$. The eccentricity $e = \sqrt{1 - b^2/a^2} = \sqrt{1 - (5/9)/(5/4)} = \sqrt{1 - 4/9} = \sqrt{5}/3$. The length of the major axis is $2a = 2(\sqrt{5}/2) = \sqrt{5}$. The centre is the intersection of $x-2y+1=0$ and $2x+y+2=0$,which is $(-4/5, -2/5)$.

For the ellipse $4(x-2y+1)^2 + 9(2x+y+2)^2 = 25$,which of the following is true?

Similar Questions

Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 5, 0)$,foci $(\pm 4, 0)$.

The longest distance of the point $(a, 0)$ from the curve $2x^2+y^2=2x$ is

The value of $k$ such that the line $y=2x+k$ touches the ellipse $3x^2+5y^2=15$ is

If the lines joining the foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b$) to an extremity of its minor axis are inclined at an angle of $60^{\circ}$ to each other, then the eccentricity of the ellipse is:

Find the number of points on the ellipse $\frac{x^{2}}{50} + \frac{y^{2}}{20} = 1$ from which a pair of perpendicular tangents can be drawn to the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module