Equation of Normal at Given Point

Q. The length of normal chord of parabola $y^2=4x$, which subtends an angle of ${90}^{\circ }$ at the vertex is

1. $8\sqrt{3}\text{units}$
2. $6\sqrt{3}\text{units}$
3. $7\sqrt{2}\text{units}$
4. $9\sqrt{2}\text{units}$

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 $\frac{1}{2}$

Q. The normal to the parabola $y^2=8x$ at the point (2, 4) meets the parabola again at the point _____.

1. (18, -12)
2. (-18, 12)
3. (18, 12)
4. (-18, -12)

Q.

The distance of the point $\left(1,2\right)$ from the line $x+y+5=0$ measured along the line parallel to $3x-y=7$ is equal to

1. $4\sqrt{10}$

2. $40$

3. $\sqrt{40}$

4. $10\sqrt{2}$

Q. If the normals of the parabola $y^2=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x-3{)}^2+(y+2{)}^2=r^2$, then the value of $r^2$ is___

Q. The tangents and normals at the ends of the latus rectum of a parabola forms a

1. square
2. parallelogram
3. rectangle
4. rhombus

Q. If the tangents and normals at the extremities of a focal chord of any standard parabola intersect at $S\equiv (x_1,y_1)$ and $R\equiv (x_2,y_2)$ respectively, then

1. If the equation of parabola is $y^2=\pm 4ax$, then $y_1=y_2$
2. If the equation of parabola is $x^2=\pm 4ay$, then $x_1=x_2$
3. the circle with focal chord as diameter will always pass through the points $R$ and $S$
4. the circle with $RS$ as diameter will always pass through extemities of focal chord

Q. Two tangents are drawn to the parabola $y^2=8x$ which meets the tangent at vertex at $P$ and $Q$ respectively. If $PQ=4\text{units}$, then the locus of the point of intersection of the two tangents is

1. $x^2=8(y+2)$
2. $x^2=8(y-2)$
3. $y^2=8(x-2)$
4. $y^2=8(x+2)$

Q. Let $AB$ is a normal chord of parabola $y^2=4x$, with foot of normal as $A(1,-2)$. If $AB=p\sqrt{q}$, where $p$ is a natural number and $q$ is a prime number, then $\frac{p}{q}$ is equal to

Q.

What angle has a tangent of $2$?

Q.

The chord AB of the parabola $y^2=4ax$ cuts the axis of the parabola at C. If $A=(a{t}_1^2,2a{t}_1),B=(a{t}_2^2,2a{t}_2)$ and AC : AB = 1:3 then

1. None of these

2. $t_2=2t_1$

3. $t_2+2t_1=0$

4. $t_1+2t_2=0$

Q. If the tangents drawn from a point to the parabola $y^2=4ax$ are also normals to the parabola $x^2=4by$, then

1. $b^2\ge 8a^2$
2. $b^2\ge 2a^2$
3. $a^2\ge 2b^2$
4. $a^2\ge 8b^2$

Q. If $PQ$ is a normal chord of the parabola $y^2=4ax$ at $P(a{t}^2,2at).$ Then the axis of the parabola divides $\overline{PQ}$ in the ratio

1. $\frac{t^2}{t^2+2}$
2. $\frac{t^2}{t^2-2}$
3. $\frac{2t^2}{t^2-2}$
4. $\frac{t^2}{2(t^2-2)}$

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. $T\in (-8,8)$
4. $T\in (-\infty ,-8)\cup (8,\infty )$

Q. Length of normal chord of parabola $y^2=4x$ which makes an inclination of $\frac{\pi }{4}$ with the positive direction of $x-$axis is

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

Q. If $y=x\sqrt{2}-8\sqrt{2}$ is a normal chord to $y^2=8x$.Then its length (in units) is

1. $12\sqrt{3}$
2. $2\sqrt{3}$
3. $16\sqrt{3}$
4. $4\sqrt{3}$

Q. If normals drawn to $y^2=12x$ makes an angle of $45°$ with $x-$axis, then foot of the normals is/are

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

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

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

Q. The normal chord of a parabola $y^2=4ax$ at a point whose ordinate is equal to abscissa, subtends a right angle at

1. focus
2. vertex
3. ends of the latusrectum
4. any point on directrix

Q. If the normal at point $(1,2)$ on the parabola $y^2=4x$ meets the parabola again a point $(t^2,2t),$ then the value of $t$ is

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

Q. The normal at $P(2,4)$ to $y^2=8x$ meets the parabola at $Q.$ Then the equation of the circle having normal chord $PQ$ as diameter is

1. $x^2+y^2-20x+8y-12=0$
2. $x^2+y^2-10x+4y-8=0$
3. $x^2+y^2-12x+6y-15=0$
4. $x^2+y^2-10x+8y-12=0$

Q. Locus of a point through which three normals of parabola $y^2=4ax$ are passing, two of which are making angles $\alpha$ and $\beta$ with positive $x-$ axis and $\tan \alpha \cdot \tan \beta =2$ is

1. $y(y^2-2ax)=0$
2. $y(y^2+2ax)=0$
3. $y(y^2-4ax)=0$
4. $y(y^2-ax)=0$

Q. Let $PQ$ be a chord of the parabola $y^2=8x$. A circle drawn with $PQ$ as diameter passes through the vertex $V$ of the parabola. If area of $\Delta PVQ=80$ square unit, then the coordinates of $P$ are

1. $(32,-16)$
2. $(-32,16)$
3. $(-32,-16)$
4. $(32,16)$

Q. The locus of the mid-points of the portion of the normal to the parabola $y^2=16x$ intercepted between the curve and the axis is another parabola whose latus rectum is

Q. If $P$ and $Q$ are the points of contact of tangents drawn from the point $T$ to $y^2=4ax$ and $PQ$ be a normal of the parabola at $P$, then the locus of the point which bisects $TP$ is

1. $x+a=0$
2. $x=1$
3. $x+2a=0$
4. $x=0$

Q.

Find the equation of normal to the parabola
$y^2=16x$ at point (4, 8)

1. x + y + 4 = 0

2. x + y - 2 = 0

3. x + y - 3 = 0

4. x + y - 4 = 0

Q. If $ax+by=1$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ then the value of $a^2–b^2$ is

1. $b^2{e}^2$
2. $\frac{1}{b^2{e}^2}$
3. $a^2{e}^2$
4. $\frac{1}{a^2{e}^2}$

Q. If the minimum distance between the parabolas $y^2-4x-8y+40=0$ and $x^2-8x-4y+40=0$ is $d$, then the value of $d^2$ is

Q. If a point $P$ on the axis of the parabola $y^2=4x$ is taken such that the point is at shortest distance from the circle $x^2+y^2+2x-2\sqrt{2}y+2=0$.Common tangents are drawn to the circle and the parabola from $P$.If the area of the $\Delta PAB$ is $\sqrt{a}$ sq. units where $A$ and $B$ are the points of contact on two distinct tangents from $P$ on circle and parabola respectively, then $a=$

Q.

Obtain the equation of hyperbola whose equations of asymptotes are x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and hyperbola passes through (1, -1).

1. 

2. 

3. 

4.