Focal Chord of Ellipse

Q.

If $P$ is any point on the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$and $S$ and $S$$$ are the foci, then $\mathrm{PS}+{\mathrm{PS}}^{}=$

1. $4$

2. $8$

3. $10$

4. $12$

Q.

The distance of the point $\left(-2,3\right)$ from the line $x-y=5$ is

1. $5\sqrt{2}$

2. $2\sqrt{5}$

3. $3\sqrt{5}$

4. $5\sqrt{3}$

Q.

If the distance of the point $P(1,-2,1)$ from the plane $x+2y-2z=\alpha$, where $\alpha >0$, is $5$, then the foot of the perpendicular from $P$ to the plane is

1. $(\frac{8}{3},\frac{4}{3}–\frac{7}{3})$

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

3. $(\frac{1}{3},\frac{2}{3},\frac{10}{3})$

4. $(\frac{2}{3},-\frac{1}{3},\frac{5}{2})$

Q.

If the distance of $p(x,y)$ from $a(5,1)$ and $b(-1,5)$ are equal, prove that $3x=2y$.

Q.

The straight lines $y=\pm x$ intersect the parabola $y^2=8x$ in points $P$ and $Q,$ then length of $PQ$ is

1. $4$

2. $4\sqrt{2}$

3. $8$

4. $16$

Q. The sum of the focal distances of any point on the ellipse $9x^2+16y^2=144$ is

1. 32
   
2. 18
   
3. 16
   
4. 8

Q. The eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ is

1. $\frac{4}{5}$
2. $\frac{3}{5}$
3. $\frac{2}{5}$
4. $\frac{1}{5}$

Q.

A point moves so that the square of its distance from the point $(3,-2)$ is numerically equal to its distance from the line $5x-12y=13$.

The equation of the locus of the point is

1. $13x^2+13y^2–83x+64y+182=0$

2. $x^2+y^2–11x+16y+26=0$

3. $x^2+y^2–11x+16y=0$

4. None of these

Q.

PSQ is a focal chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}$ = 1 then $\frac{1}{SP}+\frac{1}{SQ}$ =

1. 2/3

2. 3/2

3. 4/3

4. 4/9

Q.

The distance between $(2,3)$ and $(-4,5)$ is _______

Q. If x, {x} and 2[x] represent the segments of a focal chord and length of latus rectum of an ellipse respectively, then length of major axis of ellipse is always greater than

1. 7
2. 5
3. 8
4. 2

Q. PSP’ is a focal chord of the ellipse $16x^2+25y^2=400.$ If SP = 8 then SP’ =

1. 
2. 
3. 
4. 

Q.

The line $x=2y$ intersects the ellipse $\frac{x^2}{4}+y^2=1$ at the points $P$ and $Q$. The equation of the circle with $PQ$ as diameter is

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

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

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

4. $x^2+y^2=\frac{5}{2}$

Q. The eccentric angles of extremities of a focal chord (other than Major axis) of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are ${\theta }_1$ and ${\theta }_2$.
If the eccentricity of the ellipse are $e_1$ and $e_2$ for the conditions $a>b$ and $b>a$ respectively, then ${\cos }^2\left(\frac{{\theta }_1-{\theta }_2}{2}\right)(\frac{1}{e_1^2}+\frac{1}{e_2^2})$ is

Q.

$△ABC$ with vertices$A(-2,0),B(2,0)$ and$C(0,2)$ is similar to $△DEF$ with vertices $D(-4,0)E(4,0)$ and $F(0,4)$. State whether the following statement is true or false. Justify your answer.

1. True
2. False

Q.

If $I$ denotes the semi - latus rectum of the parabola$y^2=4ax$ and $SP$ and $SQ$ denote the segments of any focal chord $PQ,S$ being the focus, then$SP,I$ and $SQ$ are in the relation.

1. $AP$

2. $GP$

3. $HP$

4. $I^2=S{P}^2+S{Q}^2$

Q. Find the foci of the curve $16x^2-24xy+9y^2+28x+14y+21=0$.

Q. The foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ coincide. Then , the value of $b^2$ is :

1. $5$
2. $7$
3. $9$
4. $1$

Q. PSP’ is a focal chord of the ellipse $16x^2+25y^2=400.$ If SP = 8 then SP’ =

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

Q. The sum of the distances of any point on the ellipse $3x^2+4y^2=24$ from its foci is :

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

Q. If PSQ is a focal chord of the ellipse $16x^2+25y^2=400$ such that SP = 8 then the length of SQ =

1. 2
2. 
3. 16
4. 25

Q. Find the equation of the ellipse whose focus is $(1,-1)$ eccentricity is $\frac{2}{3}$ and directrix is $x+y+2=0$.

Q. If $PSQ$ is the focal chord of the parabola $y^2=8x$ such that $SP=6$. Then the length of $SQ$ is?

1. $6$
2. $4$
3. $3$
4. None of these

Q. If $PS{P}^{\prime }$ is a focal chord of the parabola $y^2=4ax,$ and $SL$ is its semi latus rectum then, $SP,SL,S{P}^{\prime }$ are in

1. A.P
2. G.P
3. H.P
4. A.G.P

Q. The sum of the focal distances of a point on the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is:

1. $4$ units
2. $6$ units
3. $10$ units
4. $8$ units

Q. The equation of ellipse with the length of major and minor axis as $12,16$ cm respectively is

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

1. $12$
2. $10$
3. $6$
4. $8$

Q. The latusrectum of the parabola $y^2=4ax$ whose focal chord is $PSQ$ such that $SP=3$ and $SQ=2$ being focus, is given by

1. $\frac{12}{5}$
2. $\frac{6}{5}$
3. $\frac{1}{5}$
4. $\frac{24}{5}$

Q. If $F_1=(3,0)$, $F_2=(-3,0)$ and $P$ is any point on the curve $16x^2+25y^2=400$, then $P{F}_1+P{F}_2$ equals to:

1. $8$
2. $6$
3. $10$
4. $12$

Q.

PSQ is a focal chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}$ = 1 then $\frac{1}{SP}+\frac{1}{SQ}$ =

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

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

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

4. $\frac{4}{9}$