Definition of Ellipse

Q. If the equation $(5x-1{)}^2+(5y-2{)}^2=({\lambda }^2-2\lambda +1)(3x+4y-1{)}^2$ represents an ellipse, then

1. $\lambda \in (0,2)-\left\{1\right\}$
2. $\lambda \in (0,2)$
3. $\lambda \in (0,1)$
4. $\lambda \in (0,1]$

Q. A ladder $12$ units long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor along $x-$axis. Then the locus of a point on the ladder $4$ units from its foot, is

1. $\frac{x^2}{32}+\frac{y^2}{32}=1$
2. $\frac{x^2}{64}+\frac{y^2}{32}=1$
3. $\frac{x^2}{8}+\frac{y^2}{4}=1$
4. $\frac{x^2}{64}+\frac{y^2}{16}=1$

Q. $\mathrm{If}P(x,y)\mathrm{is}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{ellipse}$$16x^2+25y^2=400$$\mathrm{and}$$F_1=(3,0),F_2=(-3,0),\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}{\mathrm{PF}}_1+{\mathrm{PF}}_2\mathrm{is}$

1. 2
2. 5
3. 10
4. 20

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

1. $\lambda \in (-\infty ,2)\cup (5,\infty )$
2. $\lambda \in (2,5)$
3. $\lambda \in (2,\frac{7}{2})\cup (\frac{7}{2},5)$
4. $\lambda \in (2,\frac{7}{2})\cup (5,\infty )$

Q. The equation of reflection of the ellipse $\frac{(x-4{)}^2}{16}+\frac{(y-3{)}^2}{9}=1$ about the line $x-y-2=0$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $\frac{(x-2{)}^2}{16}+\frac{(y-5{)}^2}{9}=1$
2. $\frac{(x-2{)}^2}{9}+\frac{(y-5{)}^2}{16}=1$
3. $\frac{(x-5{)}^2}{16}+\frac{(y-2{)}^2}{9}=1$
4. $\frac{(x-5{)}^2}{9}+\frac{(y-2{)}^2}{16}=1$

Q. Equation of the ellipse with focus $(3,-2)$,
eccentricity $\frac{3}{4}$ and directrix $2x-y+3=0$ is

1. $44x^2+36xy+71y^2-374x-528y+756=0$
2. $44x^2+36xy+71y^2-588x+374y+959=0$
3. $44x^2+36xy+71y^2-125x-274y+659=0$
4. $44x^2+36xy+71y^2-135x-47y+859=0$

Q. If a straight line through the point $P(\lambda ,2)$ where $\lambda \ne 0$, meets the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ at $A$ and $D$ and meets the coordinate axis at $B$ and $C$ such that $PA\cdot PD=PB\cdot PC,$ then range of $\lambda$ is
(correct answer + 2, wrong answer - 0.50)

1. $(-\infty ,-6]\cup [6,\infty )$
2. $(-\infty ,-12]\cup [12,\infty )$
3. $[6,\infty )$
4. $[12,\infty )$

Q. A ladder $12$ units long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor along $x-$axis. Then the locus of a point on the ladder $4$ units from its foot, is

1. $\frac{x^2}{8}+\frac{y^2}{4}=1$
2. $\frac{x^2}{64}+\frac{y^2}{32}=1$
3. $\frac{x^2}{32}+\frac{y^2}{32}=1$
4. $\frac{x^2}{64}+\frac{y^2}{16}=1$

Q. From any point $P$ lying in the first quadrant on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1,PN$ is drawn perpendicular to the major axis and produced at $Q$ so that $NQ$ equals to $P{S}^{\prime },$ where $S^{\prime }$ can be any foci. Then the locus of $Q$ can be

1. $3x-5y+25=0$
2. $3x+5y+25=0$
3. $3x+5y=25$
4. $3x-5y=25$

Q. If the length of the major axis of the ellipse $(5x-10{)}^2+(5y+15{)}^2=\frac{(3x-4y+7{)}^2}{4},$ is $k$ units, then $3k=$