General Equation of Ellipse

Q.

Consider an ellipse whose focus is at $(ae,0)$ and directrix along $(x=\frac{a}{e})$.
Find the equation of the ellipse if
$b^2=a^2(1-e^2)$

1. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

2. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

3. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=0$

4. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=-1$

Q. The equation of the ellipse with its focus at (6, 2), centre at (1, 2) and which passes through the point (4, 6) is

1. 
2. 
3. 
4. 

Q.

Foci of the Ellipse $25x^2+9y^2-150x-90y+225$ = 0 are

1. (5, 7) (4, 7)

2. (6, 9) (5, 1)

3. (3, 9) (3, 1)

4. (4, 9) (4, 1)

Q. Two circles are given such that one is completely lying inside the other without touching and their centers are not coincides. Then the locus of variable circle which touches the smaller circle externally and bigger circle internally, is

1. Circle
2. Parabola
3. Hyperbola
4. Ellipse

Q. The equation $3x^2+4y^2-18x+16y+43=k$

1. represents empty set, if $k<0$
2. represents an ellipse, if $k>0$
3. represents a point, if $k=0$
4. cannot represent a real pair of straight lines for any value of $k$

Q. A rod $AB$ of the length $30$ units slips on the coordinate axes such that $A$ lies on $x-$axis and $B$ lies on $y-$axis. Then, the locus of $C$ if $BC=2AC$ is

1. $4x^2+y^2=400$
2. $x^2+4y^2=100$
3. $4x^2+y^2=100$
4. $x^2+4y^2=400$

Q. The ellipse $x^2+4y^2=4$ is inscribed in a rectangle touches its side and aligned with the coordinates axes, which is turn in inscribed in another ellipse which passes through that passes through the point $(4,0).$ Then , the equation of ellipse is

1. $x^2+12y^2=16$
2. $4x^2+48y^2=48$
3. $4x^2+64y^2=48$
4. $x^2+16y^2=16$

Q. The number of integral points which lies inside the ellipse whose vertices and foci are $(\pm 5,0)$ and $(\pm 4,0)$ respectively is

Q. An ellipse, with foci at $(0,2)$ and $(0,–2)$ and minor axis of length $4$ units, passes through which of the following points?

1. $(2,\sqrt{2})$
2. $(1,2\sqrt{2})$
3. $(\sqrt{2},2)$
4. $(2,2\sqrt{2})$

Q. If the curve $x^2+2y^2=2$ intersects the line $x+y=1$ at two points $P$ and $Q,$ then the angle subtended by the line segment $PQ$ at the origin is:

1. $\frac{\pi }{2}+{\tan }^{-1}\left(\frac{1}{4}\right)$
2. $\frac{\pi }{2}-{\tan }^{-1}\left(\frac{1}{4}\right)$
3. $\frac{\pi }{2}+{\tan }^{-1}\left(\frac{1}{3}\right)$
4. $\frac{\pi }{2}-{\tan }^{-1}\left(\frac{1}{3}\right)$