Rectangular Hyperbola
Trending Questions
Q. Which of the following rectangular hyperbola has eccentricity equal to √2
- xy=100
- xy=98
- xy=0.8
- xy=1
Q. The equation of common tangent of hyperbola 9x2−9y2=8 and y2=32x is/are
- 9x+3y−8=0
- 9x−3y+8=0
- 9x+3y+8=0
- 9x−3y−8=0
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
- xy=2x+y−2
- 2xy=x+2y+5
- xy=x+y+1
- None of these
Q.
How to change standard form to vertex form of a parabola?
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×RT=
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
- xy−1=x−y
- xy+1=x+y
- 2xy−x+y=0
- 2xy+x−y=0
Q. If the tangent and normal to a rectangular hyporbola xy=c2 at a point cuts off intercepts a1 and a2 on x−axis and b1, and b2 on the y−axis, then a1a2+b1b2=
- c2
- 0
- 2c2
- 2c2
Q. A hyperbola with the assymptotes 3x + 4y = 2 and 4x - 3y = 2 is a rectangular hyperbola.
- False
- True
Q. If the curve xy=R2−16 represents a rectangular hyperbola whose branches lie only in the quadrant in which abscissa and ordinate are opposite in sign, then
- |R|<4
- |R|≥4
- |R|=4
- |R|=5
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).
- xy=4
- xy=14
- xy=1
- xy=−4
Q. If normal to the rectangular hyperbola xy=4 at the point t1(=4) meets the curve again at the point t2, then 1|t2|=
Q. The eccentricity of the rectangular hyperbola is
- 2
- √2
- \N
- None of these
Q. The length of Transverse axis and Conjugate axis is always equal in rectangular hyperbola and its eccentricity is always √2.
- True
- False
Q. The normal to curve xy=4 at the point (1, 4) meets the curve again at point
- (−4, −1)
- (−8, −12)
- (−16, −14)
- (−1, −4)
Q. Let C be a curve which is locus of the point of the intersection of lines x=2+m and my=4−m. A circle S≡(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 C, then
- eccentricity of C is √3
- eccentricity of C is √2
- OA2+OB2+OC2+OD2=120
- OA2+OB2+OC2+OD2=100
Q. Tangents are drawn from the points on a tangent of the hyperbola x2−y2=a2 to the parabola y2=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
- x2a2+y24a2=1
- x2a2+y23a2=1
- x2a2−y24a2=1
- x2a2−y23a2=1
Q. The separate equations of the asymptotes of rectangular hyperbola x2+2xycot2α−y2=a2 are :
- xcotα−y=0, xtanα+y=0
- xcotα+y=0, xtanα+y=0
- xcotα+y=0, xtanα−y=0
- xcotα−y=0, xtanα−y=0
Q. Circles are drawn on chords of the rectanglar hyperbola xy=4 parallel to the line y=x as diameters. All such circles pass through two fixed points whose coordinates are
- (2, 2)
- (2, −2)
- (−2, 2)
- (−2, −2)
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
- (2, 2)
- (2, −2)
- (−2, 2)
- (−2, −2)
Q. Let C be a curve which is locus of the point of the intersection of lines x=2+m and my=4−m. A circle S≡(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 C, then
- eccentricity of C is √3
- eccentricity of C is √2
- OA2+OB2+OC2+OD2=120
- OA2+OB2+OC2+OD2=100
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)
('Foot of ordinate of a point' is the foot of its perpendicular from the point to the x−axis)
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, CP2+CQ2+CR2+CS2 is equal to
r2
2r2
3r2
4r2
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 O1 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 OO1 , then :
- Centre of hyperbola is (0, 3)
- One of the asymptotes of the hyperbola is y=x+3
- Centre of hyperbola is (0, 4)
- One of the asymptotes of the hyperbola is y=x+4
Q. Normal at (5, 3) of rectangular hyperbola xy−y−2x−2=0 intersects it again at a point
- (34, −14)
- (−1, 0)
- (−1, 1)
- (0, −2)
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
- equation of directrices of conic are x+y=±4
- eccentricity of conic =√2
- the foci of the conic are (4, 4) and (−4, −4)
- tangent equation at (2, 4) to conic is 4x+2y=16
Q. If the sum of the slopes of the normal from a point P to the rectangular hyperbola xy=c2 is equal to λ(λ∈R+), then locus of P is
- x2=λc2
- y2=λc2
- xy=λc2
- x+y=λc2
Q. The locus of a point, from where pair of tangents to the rectangular hyperbola x2−y2=a2 contain an angle of 45∘, is :
- (x2+y2)2+4a2(x2−y2)=4a4
- (x2+y2)2+4a2(x2−y2)=a4
- 2(x2+y2)+4a2(x2−y2)=4a2
- (x2+y2)+a2(x2−y2)=4a2
Q. A rectangular hyperbola whose center is C, is cut by any circle of radius r in four points P, Q, R, S. Then CP2+CQ2+CR2+CS2=
- 4r2
- 2r2
- r2
- 8r2
Q. For the hyperbola xy=−16. Which of the following is/are true ?
- Length of latus rectum is 8√2 unit
- coordinates of vertices will be (−4, 4) and (4, −4)
- equation of directrices will be x+y=±4√2
- Length of transverse axis is 8√2 unit
Q. If normal to the rectangular hyperbola xy=4 at the point t1(=4) meets the curve again at the point t2, then 1|t2|=