General Equation of Ellipse

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. The equation of the ellipse with focus (-1, 1), directrix x-y+3=0 and eccentricity 1/2 is

1. $7x^2+2xy+7y^2+10x-10y-7=0$
   
2. $7x^2+2xy+7y^2+10x+10y+7=0$
   
3. none of these
4. $7x^2+7y^2+2xy-10y+10x+7=0$
   

Q.

Find the equation of an ellipse whose eccentricity is $\frac{2}{3}$, the latus-rectum is 5 and the centre is at the origin.

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

Q. Find the equation of an ellipse whose foci are at $(\pm 3,0)$ and which passes through (4, 1).

Q. Find the equation of the ellipse in the following cases:
(i) eccentricity $e=\frac{1}{2}$ and foci $(\pm 2,0)$
(ii) eccentricity $e=\frac{2}{3}$ snd length of latus-rectum=5
(iii) eccentricity $e=\frac{1}{2}$ and semi-major axis =4
(iv) eccentricity $e=\frac{1}{2}$ and major axis =12
(v) The ellipse passes through (1, 4) and (-6, 1).
(vi) Vertices $(\pm 5,0)$, foci $(\pm 4,0)$
(vii) Vertices $(0,\pm 13)$, foci $(0,\pm 5)$
(viii) Vertices $(\pm 6,0)$, foci $(\pm 4,0)$
(ix) Ends of major axis $(\pm 3,0)$, ends of minor axis $(0,\pm 2)$
(x) Ends of major axis $(0,\pm \sqrt{5})$, ends of minor axis $(\pm 1,0)$
(xi) Length of major axis 26, foci $(\pm 5,0)$
(xii) Length of minor axis 16, foci $(0,\pm 6)$
(xiii) Foci $(\pm 3,0)$, a=4

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.

In $\left(-4,4\right)$, the function $f\left(x\right)={\int }_{-10}^x\left(t^4-4\right)e^{-4t}dt$ has

1. No extrema

2. One extremum

3. Two extrema

4. Four extrema

Q. The point $P$ is on the $y$-axis. If $P$ is equidistant from $(1,2,3)$ and $(2,3,4),$ then $P_y=$

1. $\frac{15}{2}$
2. $\frac{3}{2}$
3. $1$
4. $3$

Q.

The equation of the parabola whose focus is $(3,-4)$ and directrix$6x-7y+5=0$, is

1. ${(7x+6y)}^2–570x+750y+2100=0$

2. ${(7x+6y)}^2+570x-750y+2100=0$

3. ${(7x-6y)}^2–570x+750y+2100=0$

4. ${(7x-6y)}^2+570x-750y+2100=0$

Q.

There are $n$ distinct points on the circumference of a circle. The number of pentagons that can be formed with these points as vertices is equal to the number of possible triangles. Then the value of $n$ is

1. $7$

2. $8$

3. $15$

4. $30$

Q. The area bounded by $xy=1,x=1,x=e$ and $x$ axis is

1. $1$
2. $0.5$
3. $2$
4. $3$

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.

Find the equation of an ellipse, the distance between the foci is 8 units and the distance between the directions is 18 units.

Q.

Find the equation to the ellipse (referred to its axes as the axes of x and y respectively) which passes through the point(-3, 1) and has eccentricity $\sqrt{\frac{2}{5}}$.

Q. Find the equation of an ellipse whose axes lie along coordinate axes and which passes through (4, 3) and (-1, 4).

Q. Find the equation of an ellipse whose vertices are $(0,\pm 10)$ and eccentricity $e=\frac{4}{5}$.

Q. Find the equation of an ellipse with its foci on y-axis, eccentricity 3/4, centre at the origin and passing through (6, 4).

Q. If $y=g\left(x\right)$ is a curve which is obtained by the reflection of $f\left(x\right)=\frac{e^x-e^{-x}}{2}$ about the line $y=x,$ then which of the following is CORRECT?

1. $g\left(x\right)$ has more than one tangent parallel to $x-$axis.
2. $g\left(x\right)$ has more than one tangent parallel to $y-$axis.
3. $y=-x$ is a tangent to $g\left(x\right)$ at $(0,0).$
4. $g\left(x\right)$ has no extremum.

Q.

Find the equation of the ellipse in the following cases:
(i) focus is (0, 1), directrix is x+y=0 and $e=\frac{1}{2}$.
(ii) focus is (-1, 1), directrix is x-y+3=0 and $e=\frac{1}{2}$.
(iii) focus is (-2, 3), directrix is 2x+3y+4=0 and $e=\frac{4}{5}$.
(iv) focus is (1, 2), directrix is 3x+4y-5=0 and $e=\frac{1}{2}$.

Q. Find the equation of an ellipse whose axes lie along the coordinate axes, which passes through the point (-3, 1) and has eccentricity equal to $\sqrt{2/5}$.

Q. If the lengths of semi-major and semi-minor axes of an ellipse are 2 and $\sqrt{3}$ and their corresponding equations are y-5=0 and x+3=0, then write the equation of the ellipse are 2 and $\sqrt{3}$ and their corresponding equations are y-5=0 and x+3=0, then write the equation of the ellipse.

Q. find the equation of the ellipse whose centre is (-2, 3) and whose semi-axis are 3 and 2 when major axis is (i) parallel to x-axis (ii) parallel to y-axis.

Q.

Find the equation of the ellipse whose focus is (1, -2), the directrix 3x-2y+5=0 and eccentricity equal to 1/2.

Q. The equation $\frac{x^2}{2-\lambda }+\frac{y^2}{\lambda -5}+1=0$ represents an ellipse, if

1. $\lambda <5$
   
2. $\lambda <2$
   
3. $2<\lambda <5$
   
4. $\lambda <2$ or < 5

Q. General solution of $(x+y{)}^2\frac{dy}{dx}=a^2,a\ne 0$ is ($c$ is arbitrary constant)

1. $\tan (x+y)=c$
2. $\frac{x}{a}=\tan \frac{y}{a}+c$
3. $\tan \frac{y+c}{a}=\frac{x+y}{a}$
4. $\tan xy=c$

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 focus of the parabola $x^2+y^2+2xy-6x-2y+3=0$ is

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

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 $k$