Rectangular Hyperbola

Q. If $P$ is a point on the rectangular hyperbola $x^2-y^2=a^2$, $C$ is its centre and $S$ and $S^{\prime }$ are foci, then $SP\cdot S^{\prime }P$ is equal to

1. $2$
2. $(CP{)}^2$
3. $(CS{)}^2$
4. $(S{S}^{\prime }{)}^2$

Q.

What will be the equation of that chord of hyperbola $25x^2-16y^2=400$, whose mid-point is (5, 3)?

1. 125x - 48y = 481

2. 127 + 33y = 341

3. 15x + 121y = 105

4. 115x - 117y = 17

Q. For the hyperbola $xy=-16$. Which of the following is/are true $?$

1. Length of latus rectum is $8\sqrt{2}$ unit
2. coordinates of vertices will be $(-4,4)$ and $(4,-4)$
3. equation of directrices will be $x+y=\pm 4\sqrt{2}$
4. Length of transverse axis is $8\sqrt{2}$ unit

Q.

The length of the line segment when tangent of the curve intersect to both axes for curve ${\left(\frac{x}{a}\right)}^{\frac{2}{3}}={\cos }^2\left(\theta \right)$ and ${\left(\frac{y}{a}\right)}^{\frac{2}{3}}={\sin }^2\left(\theta \right)$ is

1. $a$

2. $\left|a\right|$

3. $a^2$

4. $a^3$

Q. Consider a hyperbola $xy=4$ and a line $2x+y=4.$ Let the given line intersect the $x-$axis at $R.$ If a line through $R$ intersects the hyperbola at $S$ and $T,$ then the minimum value of $RS\times RT=$

Q. A tangent to a parabola $y^2=4ax$ is inclined at $\frac{\pi }{3}$ with axis of the parabola. The point of contact is

1. $(\frac{a}{3},-\frac{2a}{\sqrt{3}})$
2. $(3a,2\sqrt{3}a)$
3. $(3a,-2\sqrt{3}a)$
4. None of these

Q. Circles are drawn on chords of the rectangular hyperbola $xy=4$ parallel to the line $y=x$ as diameters. All such circles pass through two fixed points whose coordinates are

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

Q. Normal at $(5,3)$ of rectangular hyperbola $xy-y-2x-2=0$ intersects it again at a point

1. $(\frac{3}{4},-14)$
2. $(-1,0)$
3. $(-1,1)$
4. $(0,-2)$

Q. The locus of a point, from where pair of tangents to the rectangular hyperbola $x^2-y^2=a^2$ contain an angle of ${45}^{\circ }$, is :

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

Q.

Area of the triangle formed by the lines x-y=0, x+y=0 and any tangent to the hyperbola
$x^2-y^2=a^2$ is

1. $|a|$

2. $\frac{1}{2}|a|$

3. $a^2$

4. $\frac{1}{2}{a}^2$

Q. The vertices of $△ABC$ lie on a rectangular hyperbola such that the orthocentre of the triangle is $(3,2)$ and the asymptotes of the rectangular hyperbola are parallel to the coordinate axes. If two perpendicular tangents of the hyperbola intersect at the point $(1,1)$ , theb equation of the rectangular hyperbola is

1. $xy=2x+y-2$
2. $2xy=x+2y+5$
3. $xy=x+y+1$
4. None of these

Q. The vertices of $△ABC$ lie on a rectangular hyperbola such that the orthocentre of the triangle is $(3,2)$ and the asymptotes of the rectangular hyperbola are parallel to the coordinate axes. If two perpendicular tangents of the hyperbola intersect at the point $(1,1)$, then combined equation of the asymptotes is

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

Q. If a rectangular hyperbola of latus rectum $4$ units passing through $(0,0)$ have $(2,0)$ as its one focus, then equation of locus of the other focus is

1. $x^2+y^2=36$
2. $x^2+y^2=4$
3. $x^2-y^2=4$
4. $x^2-y^2=36$

Q. If $P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3)$ and $S(x_4,y_4)$ are four concyclic points on the rectangular hyperbola $xy=c^2$, then coordinates of the orthocenter of the $△PQR$ is

1. $(x_4,-y_4)$
2. $(x_4,y_4)$
3. $(-x_4,-y_4)$
4. $(-x_4,y_4)$

Q. What is the transverse axis of a hyperbola?

Q. The separate equations of the asymptotes of rectangular hyperbola $x^2+2xy\cot 2\alpha -y^2=a^2$ are :

1. $x\cot \alpha -y=0,x\tan \alpha +y=0$
2. $x\cot \alpha +y=0,x\tan \alpha +y=0$
3. $x\cot \alpha +y=0,x\tan \alpha -y=0$
4. $x\cot \alpha -y=0,x\tan \alpha -y=0$

Q.

A rectangular hyperbola whose centre is C, is cut by any circle of radius r in four points P, Q, R and S. Then, $C{P}^2+C{Q}^2+C{R}^2+C{S}^2$ is equal to

1. $r^2$

2. $2r^2$

3. $3r^2$

4. $4r^2$

Q. If tangents $OQ$ and $OR$ are drawn to variable circles having radius $r$ and the centre lying on the rectangular hyperbola $xy=1,$ then locus of circumcentre of triangle $OQR$ is $(O$ being the origin$).$

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

Q. $PM$ and $PN$ are the perpendiculars from any point $P$ on the rectangular hyperbola $xy=8$ to the asymptotes. If the locus of the mid point of MN is a conic, then the least distance of $(1,1)$ to director circle of the conic is

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

Q. A variable straight line of slope $4$ intersects the hyperbola $xy=1$ at two points. The locus of the point which divides the line segment between these two points in the $1:2$ is

1. $16x^2-y^2+10xy=2$
2. $8x^2+y^2+2xy=2$
3. $16x^2+y^2+10xy=2$
4. $8x^2-y^2+2xy=2$

Q. The coordinates of a point are $a\tan (\theta +\alpha )$ and $b\tan (\theta +\beta )$, where $\theta$ is variable, then locus of the point is

1. hyperbola
2. rectangular hyperbola
3. ellipse
4. None of the above

Q. For the hyperbola $x^2-y^2-2x-2y=4$, which of the following is/are true

1. length of conjuage axis is $4$ unit
2. centre is $(1,-1)$
3. eccentricity is $\sqrt{2}$
4. length is latus rectum is $4$ unit

Q. A variable circle whose centre lies on $y^2-36=0$ cuts rectangular hyperbola $xy=16$ at $(4t_i,\frac{4}{t_i}),i=1,2,3,4$ then $\sum _{i=1}^4\frac{1}{t_i}$ can be

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

Q. $ABCD$ is a square with $A=(-4,0),B=(4,0)$ and other vertices of the square lie above the $x-$axis. Let $O$ be the origin and $O_1$ be the mid point of $CD$. For a rectangular hyperbola if one branch passes through the points $C,D$, other branch passes through origin and its transverse axis is along the straight line $O{O}_1$ , then :

1. Centre of hyperbola is $(0,3)$
2. One of the asymptotes of the hyperbola is $y=x+3$
3. Centre of hyperbola is $(0,4)$
4. One of the asymptotes of the hyperbola is $y=x+4$