Major Axis of Ellipse

Q.

Given $\text{15 cot A = 8}$. Find the value of $\sin A$ and $secA$.

Q.

The radius of the circle passing through the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and having its centre $(0,3)$ is

1. $4$

2. $\frac{3}{7}$

3. $\sqrt{12}$

4. $\frac{7}{2}$

Q. Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ $a^2>b^2,$ passes through $(\sqrt{\frac{3}{2}},1)$, and has eccentricity$\frac{1}{\sqrt{3}}$. If a circle, centered at focus $F(\alpha ,0),$ $(\alpha >0),$ of $E$ and radius $\frac{2}{\sqrt{3}}$, intersects $E$ at two points $P$ and $Q,$ then $P{Q}^2$ is equal to

1. $3$
2. $\frac{16}{3}$
3. $\frac{8}{3}$
4. $\frac{4}{3}$

Q.

For an ellipse with eccentricity $=\frac{1}{2}$ the center is at the origin. If one directrix is $x=4$, then the equation of the ellipse is

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

2. $3x^2+4y^{2^{}}=12$

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

4. $4x^2+3y^2=12$

Q. Three normals to the parabola $y^2=x$ are drawn through a point (C, 0), then

1. None of these
2. $C=\frac{1}{4}$
3. $C=\frac{1}{2}$
4. $C>\frac{1}{2}$

Q.

If $P=(x,y),F_1=(3,0),F_2=(-3,0)$ and $16x^2+25y^2=400$, then $P{F}_1+P{F}_2$ equals

1. 8

2. 6

3. 10

4. 12

Q. A hyperbola whose transverse axis is along the major axis of the conic, $\frac{x^2}{3}+\frac{y^2}{4}=4$ and has vertices at the foci of this conic. If the eccentricity of the hyperbola is $\frac{3}{2}$, then which of the following points does NOT lie on it?

1. $(5,2\sqrt{3})$
2. $(0,2)$
3. $(\sqrt{5},2\sqrt{2})$
4. $(\sqrt{1}0,2\sqrt{3})$

Q. If (2, 4) and (10, 10) are the ends of a latus-rectum of an ellipse with eccentricity 1/2, then the length of semi-major axis is

1. none of these
2. 40/3
   
3. 20/3
   
4. 15/3
   

Q.

How do you find the foci of an ellipse not centered at the origin $?$

Q. The tangent at a point P on $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ cuts one of its directrices in Q. Then PQ subtends at the corresponding focus an angle of

1. 
2. 
3. 
4. 

Q.

Point $Q$is symmetric to $P(4,-1)$ with respect to the bisector of the first quadrant. The length of $PQ$ is:

1. $3\sqrt{2}$

2. $5\sqrt{2}$

3. $7\sqrt{2}$

4. $9\sqrt{2}$

Q. A hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3x^2+4y^2=12$, then this hyperbola does not pass through which of the following points

1. $(\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}})$
2. $(1,-\frac{1}{\sqrt{2}})$
3. $(\frac{1}{\sqrt{2}},0)$
4. $(-\sqrt{\frac{3}{2}},1)$

Q. Let $S,S^{\prime }$ be the foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose eccentricity is '$e$'. $P$ is a variable point on the ellipse. Consider the locus of the incentre of the $△PS{S}^{\prime }$

Maximum area of rectangle inscribed in the locus is

1. $\frac{2abe}{1-e}$
2. $\frac{abe}{1+e}$
3. None of these
4. $\frac{2ab{e}^2}{1+e}$

Q.

One focus of an Ellipse is (1, 0) with centre (0, 0). If the length of major axis is 6, its e =

1. 1/4

2. 2/3

3. 1/3

4. 1/2

Q.

Equation of the ellipse with foci ($\pm$5, 0) and length of major axis 26 is

1. 

2. 

3. 

4. 

Q. The focus of a conic is the origin and its corresponding directrix is $7x-y-10=0$. If length of its latus rectum is $2$, then its eccentricity is:

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

Q. $E_1=\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0,(a>b)$ and $E_2=\frac{x^2}{k^2}+\frac{y^2}{b^2}-1=0,(k<b)E_2$ is inscribed in $E_1$. If $E_1$ and $E_2$ have same eccentricities then length of minor axis of $E_2=p($LLR of $E_1)$ then $p=?$

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

Q.

If the axes of the ellipse are coordinate axes and A and B are the ends of major axis and minor axis respectively.If the area of $△$OAB is 16 sq units, e = $\frac{\sqrt{3}}{2}$ then the equation of the ellipse is

1. 

2. 

3. 

4. 

Q. LR of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a<b)$ is half of its major axis. If its eccentricity is $e=\frac{1}{\sqrt{k}}$, then value of $k$ is

Q. Find the eccentricity of an ellipse whose latus-rectum is
$\left(i\right)$ half of its minor axis

Q.

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.

Q. Write the eccentricity of an ellipse whose latus-rectum is one half of the minor axis.

Q. Find the equation of the parabola whose axis is parallel to $y$ - axis and passes through the points $(0,4),(1,9)$ and $(-2,6).$

1. ${(x+\frac{3}{4})}^2=4\frac{1}{8}(y-\frac{23}{8})$
2. ${(x-\frac{3}{4})}^2=2\frac{1}{8}(y-\frac{23}{8})$
3. ${(x+\frac{4}{3})}^2=3\frac{1}{8}(y+\frac{23}{8})$
4. ${(x+\frac{4}{3})}^2=4\frac{1}{8}(y-\frac{23}{8})$

Q. In an ellipse the length of major axis is $10$ and the distance between the foci is $8$. Then the length of minor axis is:

1. $5$
2. $7$
3. $6$
4. $4$

Q. If $ABCDEF$ is a regular hexagon, show that $\overline{AB}+\overline{AC}+\overline{AD}+\overline{AE}+\overline{AF}=6\overline{AO}$, where $O$ is the centre of the hexagon.

Q. $\frac{C_0}{1}-\frac{C_1}{4}+\frac{C_2}{7}-...+{(-1)}^n\frac{C_n}{3n+1}=$_____ where $C_r$ stands for ${{}^{n}C}_r$

1. $\frac{3.n!}{\mathrm{1.4.7...}(3n+1)}$
2. $\frac{3^{n+1}.n!}{\mathrm{1.4.7...}(3n+1)}$
3. $\frac{3^n.n!}{\mathrm{1.4.7...}(3n+1)}$
4. $\frac{3^n}{\mathrm{1.4.7...}(3n+1)}$

Q.

Why do we use only X and Y as coordinate axes why can't we use A, B, C, P, Q, R etc

Q. A circle and an ellipse have centres at $(0,0)$ and the circle passes through foci $F_1$ and $F_2$ of the ellipse, such that the two curves intersect at four points. Let $P$ be any one of their points of intersection. If the length of major axis of the ellipse is $17$ and area of the triangle $P{F}_1{F}_2$ is $30$, then distance between foci is

1. $11$
2. $12$
3. $13$
4. $15$

Q.

If $\frac{x^2}{f\left(4a\right)}+\frac{y^2}{f(a^2-5)}=1$ represents an ellipse with major axis as y-axis and f is a decreasing function, then

1. $a\in (-\infty ,1)$

2. $a\in (5,\infty )$

3. $a\in (1,4)$

4. $a\in (-1,5)$

Q. Find the equation for the ellipse that satisfies the given conditions: Length of minor axis \(16, foci (0, \pm 6).\)