Asymptotes

Q.

Why is it called a rectangular hyperbola?

Q. Let the double ordinate $P{P}^{\prime }$ of the hyperbola $\frac{x^2}{4}-\frac{y^2}{3}=1$ is produced both sides to meet asymptotes of hyperbola in $Q$ and $Q^{\prime }.$ The product $\left(PQ\right)\left(P{Q}^{\prime }\right)$ is equal to

Q. The product of the lengths of the perpendiculars from any point on the hyperbola $x^2-2y^2=2$ to its asymptotes is

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

Q. The equation of the hyperbola whose asymptotes are $2x-y=3$ and $3x+y=7$ and passing though the point $(1,1)$ is

1. $6x^2-xy-y^2-20x+4y+12=0$
2. $6x^2-xy-y^2-23x+4y+15=0$
3. $6x^2-xy-y^2+20x-4y-20=0$
4. $6x^2+xy+y^2-23x-4y+19=0$

Q. The product of the perpendiculars from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes is

1. 
2. 
   
   
3. 
   
   
4. 
   
   

Q. If the lines $ax+y+1=0,x+by+1=0,\text{and}x+y+c=0(a,b,c\text{being different from 1})$ are concurrent, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

1. $0$
2. $1$
3. $\frac{1}{a+b+c}$
4. $a+b+c$

Q. A hyperbola has focus at origin, its eccentricity is $\sqrt{2}$ and corresponding directrix is $x+y+1=0$. The equation of its asymptotes is/are:

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

Q. If two tangents can be drawn to the different branches of hyperbola
$\frac{x^2}{1}-\frac{y^2}{4}=1$ from the paint $(\alpha ,{\alpha }^2)$, then

1. 
   
2. 
3. 
   
4. 

Q. The equations of the assymptotes of the hyperbola $3x^2+10\mathrm{xy}+8y^2+14x+22y+7=0$ are .

1. $3x^2+10\mathrm{xy}+8y^2+14x+22y-7=0$
2. $3x^2+10\mathrm{xy}+8y^2+14x+22y=0$
3. $3x^2+10\mathrm{xy}+8y^2+14x+22y+14=0$
4. $3x^2+10\mathrm{xy}+8y^2+14x+22y+15=0$

Q. The tangent to the hyperbola $x^2-3y^2=3$ at the point $(\sqrt{3},0)$ when associated with two asymptotes constitutes

1. scalene triangle
2. an equilateral triangle
3. a triangle whose area is $\sqrt{3}$ sq. units
4. a triangle whose area is $\frac{\sqrt{3}}{2}$ sq. units

Q. The coordinates of a point common to a directrix and an asymptote of the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ are

1. $(\frac{25}{\sqrt{41}},\frac{20}{\sqrt{41}})$
2. $(\frac{25}{\sqrt{41}},-\frac{20}{\sqrt{41}})$
3. $(-\frac{25}{\sqrt{41}},\frac{24}{\sqrt{41}})$
4. $(\frac{25}{\sqrt{41}},\frac{24}{\sqrt{41}})$

Q. How to find inverse of a function.

Q. Consider a hyperbola whose centre is at origin. A 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 which of the following is/are correct?

1. Equation of asymptotes is $5xy+2x^2+2y^2=0$
2. Angle subtended by $AB$ at centre of the hyperbola is ${\sin }^{-1}\left(\frac{4}{5}\right)$
3. Equaiton of the hyperbola is $2x^2+2y^2+5xy=9$
4. Equation of the tangent to the hyperbola at $(-1,\frac{7}{2})$ is $3x+2y=4$

Q. The asymptotes of a hyperbola have center at the point $(1,2)$ and are parallel to the lines $2x+3y=0$ and $3x+2y=0$. If the hyperbola passes through the point $(5,3)$, then its equation is

1. $(2x+3y-8)(3x+2y-7)-154=0$
2. $(3x+2y-8)(2x+3y+7)-338=0$
3. $(3x+2y-8)(2x+3y-7)-156=0$
4. $(2x+3y-7)(3x+2y+8)-348=0$

Q. Find the locus of the point(t²-t+1, t²+t+1), t belongs to R.

Q. The center of a rectangular hyperbola lies on the line $y=2x$. If one of the asymptotes is $x+y+c=0$, then the other asymptote is

1. $x-y-3c=0$
2. $2x-y+c=0$
3. $x-y-c=0$
4. none of these

Q. If the asymptotes of the hyperbola $(x+y+1{)}^2-(x-y-3{)}^2=5$
cut each other at $A$ and the coordinate axis at $B$ and $C$ then radius of circle passing through the points $A,B,C$ is :

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

Q. In which of the following curves, the length of the normal is equal to the length of the radius vector

1. circle
2. parabola
3. ellipse
4. rectangular hyperbola

Q. If the chord of contact of tangents from 3 points A, B, C to the Circle $x^2+y^2=a^2$ are concurrent, then A, B, C will be

1. be concyclic
2. Be collinear
3. Form the vertices of triangle
4. None of these

Q. The equation of the hyperbola of given transverse axis whose vertex bisects the distance between the centre and the focus, is given by

1. 
2. 
3. 
4. None of these

Q. Equation of the hyperbola passing through the point $(1,-1)$ and having asymptotes $x+2y+3=0$ and $3x+4y+5=0$ is :

1. $3x^2-10xy+8y^2-14x+22y+15=0$
2. $3x^2+10xy+8y^2-14x+22y+35=0$
3. $3x^2+10xy+8y^2+14x+22y-8=0$
4. $3x^2+10xy+8y^2+14x+22y+7=0$

Q. If the foci of hyperbola lies on the line $y=x$, one asymptote is $y=2x$ and it is passing through the point $(3,4)$, then

1. Equation of hyperbola is $3x^2-xy+2y^2=47$
2. Equation of hyperbola is $2x^2-5xy+2y^2+10=0$
3. Eccentricity of hyperbola is $\frac{\sqrt{17}}{4}$
4. Eccentricity of hyperbola is $\frac{\sqrt{10}}{3}$

Q. In a hyperbola portion of tangent intercepted between asymptotes is bisected at the point of contact. Consider a hyperbola whose centre is at origin. A 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.

Equation of asymptotes are

1. $5xy+2x^2+2y^2=0$
2. $3x^2+4y^2+6xy=0$
3. $2x^2+2y^2-5xy=0$
4. none of these

Q. The area of the triangle formed by any tangent to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ and its asymptotes is :sq. units.

Q.

Which of the following statements are correct?

1. The particular kind of hyperbola in which the lengths of the transverse and conjugate axis are equal is called an equilateral hyperbola.

2. Eccentricity of equilateral hyperbola = $\sqrt{2}$
3. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=0$

4. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=-a^2$

1. only 1 & 2

2. only 1, 2 & 3

3. only 2, 3 & 4

4. All of them

Q. If $P_1$ and $P_2$ are the perpendiculars from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ on its asymptotes, then :

1. $P_1{P}_2=a^2+b^2$
2. $P_1+P_2=ab$
3. $P_1+P_2=(ab{)}^2$
4. $\frac{1}{P_1{P}_2}=\frac{1}{a^2}+\frac{1}{b^2}$

Q. A hyperbola has focus at origin, its eccentricity is $\sqrt{2}$ and corresponding directrix is $x+y+1=0$. The equation of its asymptotes is/are:

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

Q. The area (in sq. units) of the part of the circle $x^2+y^2=36$, which is outside the parabola $y^2=9x,$ is :

1. $24\pi +3\sqrt{3}$
2. $12\pi +3\sqrt{3}$
3. $12\pi -3\sqrt{3}$
4. $24\pi -3\sqrt{3}$

Q. How do you find the vertex of a parabola in standard form?

Q.

If the asymptote of the hyperbola $(x+y+1{)}^2-(x-y-3{)}^5=5$ cut each other at A and the coordinate axis at B and C then radius of circle passing through the points A, B, C is

1. 

2. 

3. 

4.