Normal

Q. The angle between the normals to the parabola $y^2=24x$ at points $(6,12)$ and $(6,-12)$ is :

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q. The equation of straight line passing through the point $(3,6)$ and cutting $y=\sqrt{x}$ orthogonally is

1. $4x+y=18$
2. $x+y=9$
3. $4x-y=6$
4. $x-y=9$

Q. If the line $ax+by=0$ touches the circle $x^2+y^2+2x+4y=0$ and is a normal to the circle $x^2+y^2-4x+2y-3=0,$ then value of $(a,b)$ is/are

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

Q. The line $lx+my+n=0$ is normal to the circle $x^2+y^2+2gx+2fy+c=0,$ if

1. $lg+mf+n=0$
2. $lg+mf-n=0$
3. $lg-mf-n=0$
4. $lg-mf+n=0$

Q. The equation of normal to the hyperbola $xy=4$ which is parallel to the line $2x-y=5$ is/are

1. $2\sqrt{2}x+\sqrt{2}y=6$
2. $-2\sqrt{2}x-\sqrt{2}y=6$
3. $-2\sqrt{2}x+\sqrt{2}y=6$
4. $2\sqrt{2}x-\sqrt{2}y=6$

Q. The locus of point of intersection of two normals drawn to the parabola $y^2=4ax$ which are at right angles is

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

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

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

Q. The equation of straight line passing through the point $(3,6)$ and cutting $y=\sqrt{x}$ orthogonally is

1. $4x+y=18$
2. $x+y=9$
3. $4x-y=6$
4. $x-y=9$

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

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

Q. The area of triangle formed by the tangent, normal drawn at $(1,\sqrt{3})$ to the circle $x^2+y^2=4$ and positive x-axis, is

1. $2\sqrt{3}$
2. $\sqrt{3}$
3. $4\sqrt{3}$
4. None of these

Q. Equation of the normal to $y^2=4x$ which is perpendicular to $x+3y+1=0$ is

1. $3x-y=6$
2. $3x-y=27$
3. $3x-y=33$
4. $3x-y=21$

Q.

If the line x-y+k=0 is a normal to $y^2=4ax$ then the value of k is

1. 4a

2. -a

3. -5a

4. -3a

Q.

Given parabola $y^2=4ax$, find the equation of Normal which will interest the normal at (8, 8) and the parabola at the same point.

1. 2x + y = 24

2. 2x - y = 24

3. x + y = 24

4. 2x + y + 24 = 0

Q. If the normal to a parabola $y^2=4ax$ at $P$ meets the curve again at $Q$ and if $PQ$ and the normal at $Q$ makes angle $\alpha$ and $\beta$, respectively with the $x$-axis then $\tan \alpha (\tan \alpha +\tan \beta )$ has the value equal to

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

Q. The line $ax+by+c=0$ is a normal to the circle $x^2+y^2=25.$ The portion of the line $ax+by+c=0$ intercepted by this circle is of length

Q. $\mathrm{if}\cos \theta =\frac{12}{13}\mathrm{and}\theta \in \left(0,\frac{\pi }{2}\right],\mathrm{then}144\left(\tan (-\theta )\times \sec (-\theta )\right]$