Parabola

Q. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^2=4ax$ is another parabola with directrix

1. $x=-a$
2. $x=-\frac{a}{2}$
3. $x=0$
4. $x=\frac{a}{2}$

Q. The equation of the parabola whose vertex and focus lie on the $x-$axis at distances $a$ and $a_1$ $(0<a<a_1)$ from the origin respectively, is

1. $y^2=4(a_1-a)x$
2. $y^2=4(a_1-a)(x-a)$
3. $y^2=4(a_1-a)(x-a_1)$
4. $\text{none of these}$

Q. The vertex of a parabola is the point $(a,b)$ and latus rectum is of the length $l.$ If the parabola is upward opening and its axis is parallel to the $y-$axis, then its equation is

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

Q. If the equation $\sqrt{(x-1{)}^2+(y-2{)}^2}=\sqrt{\lambda }|x+y+3|$ represents a parabola, then

1. $\lambda =1$
2. $\lambda =\frac{1}{3}$
3. Coordinates of focus are $(1,2)$
4. Coordinates of focus are $(1,0)$

Q. The radius of the largest circle, which passes through the focus of the parabola $y^2=4(x+y)$ and also contained in it, is

Q. The equation of the parabola whose vertex is at $(3,2)$ and whose focus is at $(5,2)$

1. $y^2-8x+4y+12=0$
2. $y^2-8x-4y+28=0$
3. ​​​​​​$y^2+8x+4y-36=0$
4. $y^2-8x+4y-28=0$

Q. If the equation $\sqrt{(x+5{)}^2+(y+2{)}^2}=\sqrt{\lambda }(x+\sqrt{3}y+3)$ represent a parabola, then $\frac{1}{\lambda }=$

Q. The length of the latusrectum of the parabola $x=10y^2+by+c$ is $\frac{1}{k}$, then $k=$

Q. The coordinates of an endpoint of the latusrectum of the parabola $(y-1{)}^2=4(x+1)$ are

1. $(0,3)$
2. $(1,2)$
3. $(0,-1)$
4. $(1,-2)$

Q. Axis of a parabola lies along $x$-axis. If its vertex and focus are at distances $2$ and $4$ respectively from the origin, on the positive $x$-axis then which of the following points does not lie on it?

1. $(4,-4)$
2. $(6,4\sqrt{2})$
3. $(5,2\sqrt{6})$
4. $(8,6)$

Q. If the circle $x^2+y^2+2\lambda x=0,\lambda \in R$ touches the parabola $y^2=4x$ externally, then

1. $\lambda >0$
2. $\lambda <0$
3. $\lambda >1$
4. $\lambda <1$

Q. If the parabola $y=a{x}^2+bx+c$, passes through the points $(0,3),(1,-4)$ and $(-1,4)$, then $c-(a+b)=$

Q. The equation of the parabola whose vertex and focus lie on the $x-$axis at distances $a$ and $a_1$ $(0<a<a_1)$ from the origin respectively, is

1. $y^2=4(a_1-a)x$
2. $y^2=4(a_1-a)(x-a)$
3. $y^2=4(a_1-a)(x-a_1)$
4. $\text{none of these}$

Q. The length of the latus-rectum of the parabola $169\left\{\right(x-1{)}^2+(y-3{)}^2\}=(5x-12y+17{)}^2$ is

1. $\frac{12}{13}$
2. $\frac{14}{13}$
3. $\frac{28}{13}$
4. $\frac{31}{13}$