Parametric Equation of Normal

Q. If the normals at two points $P$ and $Q$ of a parabola $y^2=4ax$ intersect at $R$ on the curve, then the product of ordinates of $P$ and $Q$ is

1. $4a^2$
2. $2a^2$
3. $-4a^2$
4. $8a^2$

Q. A normal is drawn to the parabola $y^2=9x$ at the point $P(4,6)$. A circle is described on $SP$ as a diameter, where $S$ is the focus. If the length of the intercept(in units) made by the circle on the normal at point $P$ is $L\text{units}$, then the value of $8L$ is (units)

Q. The common tangent to the parabola $y^2=32x$ and $x^2=108y$ intersects the coordinate axes at the points $P$ and $Q$ respectively . Then length of $PQ$ is

1. $2\sqrt{13}$
2. $5\sqrt{13}$
3. $3\sqrt{13}$
4. $6\sqrt{13}$

Q. If a chord which is normal to the parabola $y^2=4ax$ at one end subtends a right angle at the vertex, then its slope can be

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

Q. The two parabolas $y^2=4ax$ and $y^2=4c(x-b)$ cannot have a common normal, other than the axis unless, if

1. $\frac{a-c}{b}>2$
2. $\frac{b}{a-c}>2$
3. $\frac{b}{a-c}<2$
4. $\frac{b}{a}+c>2$

Q.

Find the normal to the curve $x=a\left(1+\cos \left(\theta \right)\right)$, $y=a\sin \left(\theta \right)$ at $\theta$ always passes through

1. $\left(a,a\right)$

2. $\left(0,a\right)$

3. $\left(0,0\right)$

4. $\left(a,0\right)$

Q. The locus of point $P$ when three normals drawn from it to parabola $y^2=4ax$ are such that two of them make complementary angles is

1. $y^2=a(x-a)$
2. $y^2=x-a$
3. $x^2=a(y-a)$
4. $x^2=y-a$

Q. Let $L$ be an end of the latus rectum of $y^2=4x$. If the normal at $L$ meets the curve again at $M$ and the normal at $M$ meets the curve again at $N,$ then area of $△LMN$ (in sq. units) is

1. $\frac{1280}{9}$
2. $\frac{640}{9}$
3. $\frac{320}{9}$
4. $\frac{160}{9}$

Q. If the shortest distance(in units) between the parabolas $y^2=4x$ and $y^2=2x-6$ is $d$ units, then the value of $d^2$ is

Q. The locus of middle point of the portion of the normal to $y^2=4ax$ intercepted between curve and axis of parabola is

1. $y^2=a(x-a)$
2. $y^2=2a(x-a)$
3. $y^2=(x-2a)$
4. $y^2=a^2(x-2a)$

Q.

Find the equation of normal to the parabola $y^2=4axat(a{t}^2,2at)$ in terms of t, a.

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

2. $y=-tx+a{t}^3-2at$

3. $y=-tx+a{t}^3+2at$

4. $y=-tx+a{t}^3+at$

Q. The length of the transverse axis of a hyperbola is 7 and it passes through the point (5, -2). The equation of the hyperbola is

1. 

2. None of these

3. 

4. 

Q. The point of intersection of normals to the parabola $y^2=4x$ at the points whose ordinates are $4$ and $6$ is

1. $(30,-21)$
2. $(21,-30)$
3. $(17,-19)$
4. $(19,-17)$

Q. The chord $x+y=1$ of the curve $y^2=12x$ cuts it at the points $A$ and $B.$ The normals at $A$ and $B$ intersect at $C.$ If a third line from $C$ cuts the curve normally at $D,$ then the co-ordinates of $D$ are

1. $(3,-6)$
2. $(12,-12)$
3. $(6,-3)$
4. $(12,12)$

Q.

The length of the straight line $x-3y=1$ intercepted by the hyperbola $x^2-4y^2=1$, is

1. $\sqrt{10}$

2. $\frac{6}{5}$

3. $\frac{1}{\sqrt{10}}$

4. $\frac{6}{5}\sqrt{10}$

Q. Consider a parabola $y^2=4ax.$ If the normal to the parabola at the point $(a{t}^2,2at)$ cuts the parabola again at $(a{T}^2,2aT),$ then

1. $T^2\ge 8$
2. $T^2\le 6$
3. $T\in (-\infty ,-8)\cup (8,\infty )$
4. $T\in (-8,8)$

Q.

In $∆ABC$, if $AB=5,\angle B={\cos }^{-1}\left(\frac{3}{5}\right)$ and the radius of circumcircle of triangle is $5$.

Then the area of $∆ABC$ is

1. $6+8\sqrt{3}$

2. $3+4\sqrt{3}$

3. $3+8\sqrt{3}$

4. $6+8\sqrt{3}$

Q. If a variable chord $PQ$ of the parabola $y^2=4ax$ is drawn parallel to $y=x,$ then the locus of point of intersection of normals at $P$ and $Q$ is

1. $2x-y+10a=0$
2. $2x-y-8a=0$
3. $2x-y+6a=0$
4. $2x-y-12a=0$

Q. If $|z_1|=|z_2|=|z_3|=1$ and $z_1,z_2,z_3$ are represented by the vertices of an equilateral triangle. Then

1. none of these
2. $z_1+z_2+z_3$
3. $z_1{z}_2+z_2{z}_3+z_3{z}_1=1$
4. $z_1+z_2+z_3=1$

Q. The normals at two points P and Q of a parabola $y^2=4x$ meet at $(x_1,y_1)$ on the parabola. Then $P{Q}^2$ =

1. 
2. 
3. 
4. 

Q. If $a\ne 0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay,$ then

1. $d^2+(2b+3c{)}^2=0$
2. $d^2+(3b+2c{)}^2=0$
3. $d^2+(3b-2c{)}^2=0$
4. $d^2+(2b-3c{)}^2=0$

Q. The equation of the locus of the point of intersection of two normals to the parabola $y^2=4ax$ which are perpendicular to each other is

1. 
2. 
3. 
4. 

Q. Consider a family of a circle which are passing through the point (-1, 1) and are tangent to x-axis, If (h, k) are the co-ordinate of the centre of the circles, then the set of value of K is given by the interval.

Q. The normal a point $(b{t}_1^2,2b{t}_1)$ on a parabola meets the parabola again in the point $(b{t}_2^2,2b{t}_2)$ then

1. 
2. 
3. 
4. 

Q. If a variable chord $PQ$ of the parabola $y^2=4ax$ is drawn parallel to $y=x,$ then the locus of point of intersection of normals at $P$ and $Q$ is

1. $2x-y-12a=0$
2. $2x-y+10a=0$
3. $2x-y-8a=0$
4. $2x-y+6a=0$

Q. Let $P$ and $Q$ be two distinct points on the parabola $y^2=4x$, with parameters $t$ and $t_1$ respectively. If the normals at $P$ passes through $Q,$ then the minimum value of $t_1^2$ is

Q. The minimum distance between the curves $x^2=4y$ and $x^2+y^2+18x+12y+81=0$

1. $7\sqrt{2}$
2. $7\sqrt{2}-6$
3. $7\sqrt{2}+6$
4. $8$

Q.

Find the equation of normal to the parabola $y^2=4axat(a{t}^2,2at)$ in terms of t, a.

1. 

2. 

3. 

4. 

Q. If ${(1+x+x^2)}^n=\sum _{r=0}^{2n}{a}_r{x}^r=a_0+a_{1}x+a_2{x}^2+...+a^{2n}{x}^{2n}$ and
$P=a_0+a_3+a_6+...$
$Q=a_1+a_4+a_7+...$
$R=a_2+a_5+a_8+...$
then the set of values of $P,Q,R$ are respectively equals

1. $(1,1,1)$
2. $(3^{n+1},3^{n+1},3^{n+1})$
3. $(3^{n-1},3^{n-1},3^{n-1})$
4. $(3^n,3^n,3^n)$

Q. Let $P$ be the point on the parabola $y^2=4x$, which is at the shortest distance from the centre S of the circle $x^2+y^2-4x-16y+64=0$. Let $Q$ be the point on the circle dividing the line segment SP internally. Then,

1. $SP=2\sqrt{5}$
   
2. $SQ:QP=(\sqrt{5}+1):2$
3. The $x$-intercept of the normal to the parabola at $P$ is $6$
4. The slope of the tangent to the circle at $Q$ is $\frac{1}{2}$