Position of a Point W.R.T Hyperbola

Q.

If $\frac{x^2}{36}-\frac{y^2}{k^2}=1$ is a hyperbola then which of the following statement can be true?

1. (5, 2) lies on the hyperbola

2. (-3, 1) lies on the hyperbola

3. (3, 1) lies on the hyperbola

4. (10, 4) lies on the hyperbola

Q. The coordinates of a point on the hyperbola, $\frac{x^2}{24}-\frac{y^2}{18}=1,$ which is nearest to the line $3x+2y+1=0$ are

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

Q. Find the area of the region bounded by the parabola $y=x^2$ and $y=|x|$.

Q. The equation of common tangent to the hyperbola $9x^2-16y^2=144$ and circle $x^2+y^2=9$ is

1. $y=\sqrt{\frac{6}{7}}x+\frac{15}{\sqrt{7}}$
2. $y=3\sqrt{\frac{2}{7}}x+\frac{15}{\sqrt{7}}$
3. $y=3\sqrt{\frac{2}{7}}x+15\sqrt{7}$
4. $y=\sqrt{\frac{6}{7}}x+15\sqrt{7}$

Q. The number of tangents to the hyperbola $\frac{x^2}{4}-\frac{y^2}{3}=1$ through $(1,4)$ is

Q.

The position of the point (2, 5) relative to the hyperbola $9x^2-y^2=1$

1. None of these

2. Outside

3. lie on

4. Inside

Q. If the line $5x+12y=9$ touches the hyperbola $x^2-9y^2=9$, then its point of contact is

1. $(5,-\frac{4}{3})$
2. $(5,\frac{4}{3})$
   
3. $(-3,2)$
4. $(3,0)$

Q.

A common tangent to $9x^2-16y^2=144and{x}^2+y^2=9$ is,

1. 

2. 

3. 

4. 

Q. The equation of common tangent(s) to the hyperbola $9x^2-16y^2=144$ and circle $x^2+y^2=9$ is/are

1. $\sqrt{7}y=-2\sqrt{2}x-15$
2. $\sqrt{7}y=3\sqrt{2}x+15$
3. $\sqrt{7}y=2\sqrt{2}x+15$
4. $\sqrt{7}y=-3\sqrt{2}x-15$

Q. Let $A\left(\theta \right)$ and $B\left(\varphi \right)$ are the parametric ends of a chord of the hyperbola $\frac{x^2}{144}-\frac{y^2}{25}=1$. If the equation of $AB$ is $2x+3y=1$, then

1. $\tan \left(\frac{\theta +\varphi }{2}\right)=15$
2. $23\left(\tan \frac{\theta }{2}\tan \frac{\varphi }{2}\right)=25$
3. $\tan \left(\frac{\theta +\varphi }{2}\right)=-15$
4. $25\left(\tan \frac{\theta }{2}\tan \frac{\varphi }{2}\right)=23$

Q. A hyperbola passes through the point $P(\sqrt{2},\sqrt{3})$ and has focii at $(\pm 2,0).$ Then the tangent to this hyperbola at $P$ also passes through the point:

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

Q. The position of the point $(5,-4)$ relative to the hyperbola $9y^2-x^2=1$ is :

1. inside
2. outside
3. on the hyberbola
4. on the directrix of hyperbola

Q. If $4\tan \theta =3$, evaluate $\left(\frac{4\sin \theta -\cos \theta +1}{4\sin \theta +\cos \theta -1}\right)$.

Q.

Which among the following points lies outside the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$.

1. (4, 3)

2. (9, 3)

3. (4, 3.5)

4. (7, 4)

Q. If the points $(\sqrt{3}\text{sin}\theta ,\sqrt{4}\text{cos}\theta )$ where $\theta \in R$, lies outside the hyperbola $\frac{x^2}{4}-\frac{y^2}{5}=1$, then the value of $\theta$ is:

1. $\theta \in R$
2. $\theta \in R-\left\{\frac{\pi }{4}\right\}$
3. $\theta \in R-\{\frac{\pi }{2},0\}$
4. $\theta \in R-\{\frac{\pi }{4},0,\frac{\pi }{2}\}$

Q. Let $t_1,t_2,t_3$ be three points on the parabola $y^2=4x$. If the normal at $t_1$ intersects the parabola at $t_2$ and the normal at $t_2$ intersects the parabola at $t_3$ such that $3t_1+13t_2+9t_3=0$, then

1. $t_1=1$
2. $t_1=2$
3. $t_2=3$
4. $t_3=\frac{1}{3}$

Q. Which among the following point lie inside the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$

1. $(3,1)$
2. $(5,3)$
3. $(4,5)$
4. $(1,0)$

Q. Let $f\left(x\right)=lnmx(m>0)$ and $g\left(x\right)=px$. Then the equation $|f\left(x\right)|=g\left(x\right)$ has exactly two solutions (not necessarily distinct) for

1. $p=\frac{m}{e}$
2. $p=\frac{e}{m}$
3. $0<p\le \frac{e}{m}$
4. $0<p\le \frac{m}{e}$

Q. The tangent to the curve $x=1/t,y=t$ $-1/t$ at $t=2$ has _____ x-intercept and _____ y-intercept.

1. $positive,positive$
2. $positive,negative$
3. $negative,positive$
4. $negative,negative$

Q. The range of $\theta$ for which the point $(\sqrt{3}\sin \theta ,\sqrt{4}\cos \theta )$ lies outside $\frac{x^2}{4}-\frac{y^2}{5}=1$ is

1. $\theta \in R$
2. $\theta \in R-\left\{\frac{\pi }{4}\right\}$
3. $\theta \in R-\{\frac{\pi }{2},0\}$
4. $\theta \in R-\{\frac{\pi }{4},0,\frac{\pi }{2}\}$

Q. Consider a hyperbola whose centre is at origin. If line $x+y=2$ touches this hyperbola at $P(1,1)$ and intersects the asymtotes at $A$ and $B$ such that $AB=6\sqrt{2}$ units, then the equation of the tangent to the hyperbola at $(-1,\frac{7}{2})$ is

1. $x-2y=-8$
2. $2x-y=-\frac{11}{2}$
3. $x+2y=6$
4. $3x+2y=4$

Q.

The ratio in which the sphere $x^2+y^2+z^2=504$ divides the line segment AB joining the points A(12, -4, 8) and (27, -9, 18) is given by

1. 2:3 externally

2. 2:3 internally

3. 1:2 externally

4. None of these

Q. If c>0 and the area of the region enclosed by the parabolas $y=x^2-c^2$ and $y=c^2-x^2$ is 576, then c =

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

Q. If $\overline{a}$ and $\overline{b}$ are unit vectors and $\theta$ is the angle between them, show that $\sin \frac{\theta }{2}=\frac{1}{2}|\overrightarrow{a}-\overrightarrow{b}|$.

Q. The range of $\theta$ for which the point $(\sqrt{3}\sin \theta ,\sqrt{4}\cos \theta )$ lies outside $\frac{x^2}{4}-\frac{y^2}{5}=1$ is

1. $\theta \in R$
2. $\theta \in R-\left\{\frac{\pi }{4}\right\}$
3. $\theta \in R-\{\frac{\pi }{2},0\}$
4. $\theta \in R-\{\frac{\pi }{4},0,\frac{\pi }{2}\}$

Q. A hyperbola passes through the point $P(\sqrt{2},\sqrt{3})$ and has focii at $(\pm 2,0).$ Then the tangent to this hyperbola at $P$ also passes through the point:

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

Q. Tangents are drawn to $x^2+y^2=16$ from the point $P(0,h)$. These tangents meet the x-axis at $A$ and $B$. If the area of $\Delta PAB$ is minimum, then $h=?$

1. $12\sqrt{2}$
2. $6\sqrt{2}$
3. $8\sqrt{2}$
4. $4\sqrt{2}$

Q. Let $A\left(\theta \right)$ and $B\left(\varphi \right)$ are the parametric ends of a chord of the hyperbola $\frac{x^2}{144}-\frac{y^2}{25}=1$. If the equation of $AB$ is $2x+3y=1$, then

1. $\tan \left(\frac{\theta +\varphi }{2}\right)=15$
2. $23\left(\tan \frac{\theta }{2}\tan \frac{\varphi }{2}\right)=25$
3. $\tan \left(\frac{\theta +\varphi }{2}\right)=-15$
4. $25\left(\tan \frac{\theta }{2}\tan \frac{\varphi }{2}\right)=23$

Q. The equation of the tangent to the parabola $y^2=8x$ is $y=x+2$ . The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

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

Q. The tangent and normal at the point $P(a{t}^2,2at)$ to the parabola $y^2=4ax$ meet the $x$-axis at $T$ and $G$ respectively. Then the angle at which the tangent at $P$ to the parabola is inclined to the tangent at $P$ to the circle through $T,P,G$ is

1. ${\cot }^{-1}{t}^2$
2. ${\tan }^{-1}{t}^2$
3. ${\cot }^{-1}t$
4. ${\tan }^{-1}\pm t$