Asymptotes

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

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. 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 rectangular hyperbola is represented by the equation $x^2-y^2=2^2$, calculate the eccentricity value.

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

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

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. The equation of the hyperbola which passes through the origin and having the assymptotes as 3x - 4y + 7 = 0 and 4x + 3y + 1 = 0 is .

1. $12x^2-7\mathrm{xy}-12y^2-31x+17y=0$
   
2. $12x^2-7\mathrm{xy}-12y^2+31x-17y=0$
   
3. $12x^2-7\mathrm{xy}-12y^2-31x-17y=0$
   
4. $12x^2-7\mathrm{xy}-12y^2+31x+17y=0$

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 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. Locus of a point, whose chord of contact with respect to the circle $x^2+y^2=4$ is a tangent to hyperbola $xy=1$ is :

1. $xy=4$
2. $x^2-y^2=6$
3. $xy=8$
4. $x^2-y^2=16$

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

1. $\frac{ab}{a+b}$
2. $\frac{a^2{b}^2}{a^2+b^2}$
3. $\frac{2a^2{b}^2}{a^2+b^2}$
4. None of these

Q. If a hyperbola passes through (2, 3) and has asymptotes 3x - 4y + 5 = 0 and 12x + 5y - 40 = 0, then the equation of its transverse axis is

1. 77x - 21y - 265 = 0
2. 21x - 77y + 265 = 0
3. 21x - 77y - 265 = 0
4. 21x + 77y - 265 = 0

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. What are the equations of the asymptotes of a hyperbola
$\frac{x^2}{16}-\frac{y^2}{9}=1$

1. $y=\frac{3}{4}x$
2. $y=-\frac{4}{3}x$
3. $y=\frac{4}{3}x$
4. $y=-\frac{3}{4}x$

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