Equation of Normal When Slope Is Given

Q. If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal, then which one of the follwing is a valid choice for the ordered triplets $(a,b,c)$?

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

Q. If $2x+y+k=0$ is a normal to the parabola $y^2=-8x$, then the value of $k$ is

Q. Shortest distance (in units) between the two parabolas $y^2=x-2,x^2=y-2$ is

1. $\frac{1}{4\sqrt{2}}$
2. $\frac{5}{4\sqrt{2}}$
3. $\frac{7}{2\sqrt{2}}$
4. $\frac{6}{7\sqrt{2}}$

Q. The line y = 2x + k is a normal to the parabola $y^2=4x$ then k =

1. -12
2. 10
3. -10
4. 12

Q. If the normal at the point $P(a{p}^2,2ap)$ meets the parabola at $Q(a{q}^2,2aq)$ such that the lines joining the origin to $P$ and $Q$ are at right angle, then

1. $p^2=2$
2. $pq-4=0$
3. $pq+4=0$
4. $p^2=4$

Q. If the sum of the slopes of the normal from a point $P$ to the hyperbola $xy=c^2$ is equal to $\lambda (\lambda \in R^{+})$, then the locus of the point $P$ is

1. $x^2=\lambda c^2$
2. $y^2=\lambda c^2$
3. $xy=\lambda c^2$
4. $x^2-y^2=\lambda c^2$

Q. The equation of the line of the shortest distance between the parabola $y^2=4x$ and the circle $x^2+y^2-4x-2y+4=0$

1. x+y=3
2. x-y=3
3. 2x+y=5
4. done

Q. $F$ is the family of curves in the $XY$ plane., given by $y=x+c{e}^{-x}$, where $c\in R$ is an arbitrary constant. The orthogonal trajectories of the curve of the family $F$ forms the family $F^{\prime }$. Then which of the following statement(s) is\are TRUE for $F^{\prime }$.

1. $x{e}^y=e^y(y-2)+c$, is general solution for $F^{\prime }$
2. $x=y-2$ is a particular solution for $F^{\prime }$
3. If $F^{\prime }$ passes through $(4,0)$, then $F^{\prime }$ is $x=(y-2)+6e^y$
4. $xe=e(y-2)+c$, is general solution for $F^{\prime }$

Q. The equation of the normals at the end points of the latus rectum of the parabola $y^2=8x$ is/are

1. $x+y+6=0$
2. $x-y+6=0$
3. $x-y-6=0$
4. $x+y-6=0$

Q.

If a circle and rectangular hyperbola $xy=c^2$ meets in 4 points $P,Q,RandS$ then $O{P}^2+O{Q}^2+O{R}^2+O{S}^2=$______ where r is the radius of the circle. O is the origin.

1. 

2. 

3. 

4. 

Q. The line $x+2y=36$ is normal to the parabola $x^2=12y$ at the point whose distance from the focus of the parabola is

Q. A parabolic mirror is kept along $y^2=4x.$ A ray along $y=3$ strikes the mirror (from the positive direction of $x-$axis) and gets reflected along the line $L=0.$ Then which of the following is NOT true ?

1. The line $L=0$ passes through $(0,-\frac{12}{5}).$
2. The line $L=0$ intersects the $x-$axis at $(1,0).$
3. The line $L=0$ is a tangent to the circle $x^2+y^2=1.$
4. The line $L=0$ makes acute angle with positive $x-$axis in anticlockwise direction.

Q. Shortest distance between the line $y-x=1$ and the curve $x=y^2$ is

1. $\frac{\sqrt{3}}{4}$
2. $\frac{3\sqrt{2}}{8}$
3. $\frac{3\sqrt{2}}{5}$
4. $\frac{2\sqrt{3}}{8}$

Q. Let O be the origin and A be a point on the curve $y^2=4x$ Then the locus of the midpoint of OA is

1. 
2. 
3. 
4. 

Q. The maximum possible area bounded by the parabola $y=x^2+x+10$ and a chord of the parabola of length $1$ is.

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

Q.

The foci of the conjugate hyperbola of the hyberbola $\frac{x^2}{4}-\frac{y^2}{12}=1$

1. 

2. 

3. 

4. 

Q.

If point (h, k) lies on the axis of the parabola $y^2=4ax$, then find the condition for point (h, k) sothat exactly three normal can be drawn.

1. 

2. 

3. 

4. 

Q. The equation of normal to the curve $x^{\frac{1}{3}}+y^{\frac{1}{3}}=2$ at $(1,1)$ is

1. $y=2x$
2. $y=x$
3. $y=\frac{1}{2}x$
4. $y=4x$

Q. Find the coordinate of focus equation of directrix langth of latus Rectum and coordinate of end front of latus Rectum for the parabola.
$x^2+12y=0$

Q. The line y = 2x + k is a normal to the parabola $y^2=4x$ then k =

1. 12
2. -12
3. 10
4. -10

Q. The foot of the normal 3x + 4y =7 to the hyperbola $4x^2-3y^2=1$ is

1. 
2. 
3. 
4. 

Q. Find the coordinate of focus equation of directrix langth of latus Rectum and coordinate of end front of latus Rectum for the parabola.
$5x^2=8y$

Q. Find the equation at normal to the parabola at point $P(a{t}^2,2at)$ at $y^2=4ax$.

1. $y+tx=a{t}^2=2a{t}^3$
2. $y+atx=a{t}^2+2a{t}^3$
3. $y+tx=a{t}^3+2at$
4. $Noneofthese$

Q. If $x-1=0$ is the directix of parabola $y^2-kx+8=0$, then $k$ is equal to

1. $4$
2. $8$
3. $1/8$
4. $1/4$

Q. If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal, then which one of the follwing is a valid choice for the ordered triplets $(a,b,c)$?

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

Q. The equation of the normals at the end points of the latus rectum of the parabola $y^2=8x$ is/are

1. $x+y+6=0$
2. $x-y+6=0$
3. $x-y-6=0$
4. $x+y-6=0$

Q. If $2x+y+k=0$ is a normal to the parabola $y^2=-8x$, then the value of $k$ is

Q.

Find the equation of normal having slope 1 to the parabola
$y^2=4x$

1. $y=2x-3$

2. $y=x-3$

3. $y=x+3$

4. $y=x$

Q.

Given slope m, find the equation of normal having slope m to the parabola $y^2=4ax$

1. $y=mx+2am-a{m}^3$

2. $y=mx-2am-a{m}^3$

3. $y=mx-2am-m^3$

4. $y=mx-2am+a{m}^3$

Q. The limiting position of the point of intersection of the straight lines 3x + 5y = 1 and $(2+c)x+5c^{2}y=1$ as c tends to one is:

1. $(\frac{2}{5},-\frac{1}{25})$
2. $(\frac{1}{2},-\frac{1}{10})$
3. $(\frac{3}{8},-\frac{1}{40})$
4. none