Rectangular Hyperbola

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. The vertices of a right angled triangle lies on a rectangular hyperbola $xy=4$. The angle between the tangent at the right angled vertex and the hypotenuse of the triangle is $\frac{\alpha \pi }{12}$ , then $\alpha$ is

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. Tangents are drawn at two points of a hyperbola $xy=1$. Let tangent at one point passes through the foot of ordinate of the other point. If the locus of the point of intersection of the two tangents is the hyperbola $xy=a$, then the value of $9a$ is
('Foot of ordinate of a point' is the foot of its perpendicular from the point to the $x-$axis)

Q. The normal to curve $xy=4$ at the point $(1,4)$ meets the curve again at point

1. $(-4,-1)$
2. $(-8,-\frac{1}{2})$
3. $(-16,-\frac{1}{4})$
4. $(-1,-4)$

Q.

Equation of chord having mid-point (h, k) oblique hyperbola $xy=c^2$ is same as equation of tangent at (h, k) on the hyperbola $xy=c^2$

1. True

2. False

Q. If three distinct normals can be drawn to the parabola $y^2-2y=4x-9$ from the point (2a, b), then the least integral value of a is ___.

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.

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 $(1,1)$, 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. Normal at $(5,3)$ of rectangular hyperbola $xy-y-2x-2=0$ intersects it again at a point

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

Q. A standard equation of conic satisfies the point $(2,4)$ and the conic is such that the segment of any of its tangents at any point contained between the coordinate axes is bisected at the point of tangency. Then

1. equation of directrices of conic are $x+y=\pm 4$
2. eccentricity of conic $=\sqrt{2}$
3. the foci of the conic are $(4,4)\text{and}(-4,-4)$
4. tangent equation at $(2,4)$ to conic is $4x+2y=16$

Q. If the tangent and normal to a rectangular hyporbola $xy=c^2$ at a point cuts off intercepts $a_1$ and $a_2$ on $x-$axis and $b_1$, and $b_2$ on the $y-$axis, then $a_1{a}_2+b_1{b}_2=$

1. $c^2$
2. $\frac{2}{c^2}$
3. $0$
4. $2c^2$

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. Let ${}^{\prime }{C}^{\prime }$ be a curve which is locus of the point of the intersection of lines $x=2+m$ and $my=4-m$ . A circle $S\equiv (x-2{)}^2+(y+1{)}^2=25$ intersects the curve $C$ at four points $A,B,C$ and $D$. If $O$ is centre of the curve ${}^{\prime }{C}^{\prime }$ then

1. eccentricity of $C$ is $\sqrt{3}$
2. eccentricity of $C$ is $\sqrt{2}$
3. $O{A}^2+O{B}^2+O{C}^2+O{D}^2=120$
4. $O{A}^2+O{B}^2+O{C}^2+O{D}^2=100$

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

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

1. $4r^2$
2. $2r^2$
3. $r^2$
4. $8r^2$

Q. The area of the triangle formed by the coordinate axes and tangent to the curve $y={\log }_{e}x$ at $(1,0)$ is

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

Q. If locus of a point, whose chord of contact with respect to the circle $x^2+y^2=4$ is a tangent to the hyperbola $xy=1$ is $xy=c^2$, then value of $c^2=$

Q. If the curve $xy=R^2-16$ represents a rectangular hyperbola whose branches lie only in the quadrant in which abscissa and ordinate are opposite in sign, then

1. $|R|<4$
2. $|R|\ge 4$
3. $|R|=4$
4. $|R|=5$

Q. The length of the transverse axis of the rectangular hypeerbola $xy=18$ is unit

Q. Tangents are drawn from the points on a tangent of the hyperbola $x^2-y^2=a^2$ to the parabola $y^2=4ax$. If all the chords of contact pass through a fixed point $Q$, then the locus of the point $Q$ for different tangents on the hyperbola is

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

Q. If the curve $xy=c(c>0)$ and the circle $x^2+y^2=1$ touches at two points, then distance between their points of contacts is unit

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

Q. If the normals at $(x_i,y_i),\text{where,}i=1,2,3,4$ on the rectangular hyperbola $xy=c^2$ meet at $(\alpha ,\beta )$. and $x_1^2+x_2^2+x_3^2+x_4^2$ is $a$ and $y_1^2+y_2^2+y_3^2+y_4^2$ is $b$, then $a+b$ is

1. $\alpha +\beta$
2. $2(\alpha +\beta )$
3. ${\alpha }^2+{\beta }^2$
4. $2({\alpha }^2+{\beta }^2)$

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. Let $P(4,3)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at $P$ intersects the X-axis at $(16,0)$, then the eccentricity of the hyperbola is

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

Q. The length of Transverse axis and Conjugate axis is always equal in rectangular hyperbola and its eccentricity is always $\sqrt{2}$.

1. True
2. False

Q.

The method employed to find the width of a big river is called ____________ method.