PROBLEMS
(1)
06 - HYPERBOLA
x2
If a tangent to the hyperbola
a2
minor axis in T’, then prove that
y2
-
Page 1
= 1 intersects the major axis in T and
b2
a2
-
CT 2
b2
= 1 , where C is the centre of the
CT' 2
hyperbola.
(2)
1
tan - ( 2
(3)
Prove that the product of the lengths of the perpendicular line segments from any
x2
a2
y2
-
a 2b 2
= 1 to its asymptotes is
b2
a2 + b2
.
Find the co-ordinates of foci, equations of directrices, eccentricity and length of the
latus-rectum for the following hyperbolas:
2
2
2
2
( i ) 25x - 144y = - 3600,
( ii ) x - y = 16.
25
13 288

 Ans : ( i ) ( 0, ± 13 ), y = ± 13 , e = 5 , 5

(5)
- 2y2 = 1 is
x
2 ).
point on the hyperbola
(4)
2
Show that the angle between two asymptotes of the hyperbola
x2
If the eccentricities of the hyperbolas
then prove that
e 1-
2
+ e2-
2
( ii ) ( ± 4 2 , 0 ), x = ± 2 2 , e =
-
a2
y2

2, 8 

= ± 1 are e1 and e2 respectively,
b2
= 1.
( 6 ) Prove that the equation of the chord of the hyperbola
x2
a2
-
y2
b2
= 1 joining P ( α )
α + β
y
x
α - β
α + β
cos
sin
= cos
. If this chord passes through
a
2
b
2
2
α
β
1- e
the focus ( ae, 0 ), then prove that tan tan
=
2
2
1+ e
and Q ( β ) is
( 7 )
If
x2
a2
(8)
θ + φ = 2α
-
y2
b2
( constant ),
then prove that all the chords of the hyperbola
= 1 joining the points P ( θ ) and Q ( φ ) pass through a fixed point.
If the chord
PQ
of the hyperbola
centre C, then prove that
1
CP
2
+
x2
a2
1
CQ
2
-
=
y2
b2
1
a2
= 1
-
subtends a right angle at the
1
b2
( b > a ).
PROBLEMS
(9)
( 10 )
06 - HYPERBOLA
For a point on the hyperbola,
x2
a
2
-
y2
b
2
Page 2
= 1 , prove that SP ⋅ S' P = CP
2
Find the equation of the common tangent to the hyperbola
2
parabola y = 4x.
2
3x
- a2 + b2.
- 4y2 = 12 and
[ Ans: ± y = x + 1 ]
( 11 )
Find the condition for the line x cos α + y sin α = p to be a tangent to the hyperbola
x2
a2
-
y2
= 1.
b2
2
2
2
2
2
[ Ans: p = a cos α - b sin α ]
( 12 )
Find the equation of a common tangent to the
y2
a2
-
x2
a2 - b2 ,
x + y = ±
a2
-
y2
b2
= 1
and
a2 - b2 ]
Find the equation of the hyperbola passing through the point
asymptotes y = ± 5x.
[ Ans: 25x
x2
= 1 ( a > b ).
b2
[ Ans: x - y = ±
( 13 )
hyperbolas
2
( 1, 4 )
and having
- y2 = 9 ]
( 14 ) Prove that the area of the triangle formed by the asymptotes and any tangent of the
hyperbola
( 15 )
x2
a2
-
y2
b2
= 1 is ab.
A line passing through the focus S and parallel to an asymptote intersects the
hyperbola at the point P and the corresponding directrix at the point Q. Prove that
SQ = 2 SP.
PROBLEMS
06 - HYPERBOLA
Page 3
( 16 ) K is the foot of perpendicular to an asymptote from the focus S of the rectangular
hyperbola. Prove that the hyperbola bisects SK .
( 17 )
A line passing through a point P on the hyperbola and parallel to an asymptote
intersects the directrix in K. Prove that PK = SP.
( 18 ) If the chord of the hyperbola
x2
a2
-
y2
b2
= 1 joining the points α and β subtend
α
β
cot
= 0.
2
2
the right angle at the vertex ( a, 0 ), then prove that a 2 + b 2 cot
( 19 ) Find the condition that the line l x + my + n = 0 may be a tangent to the hyperbola
x2
a2
-
y2
b2
= 1 and find the co-ordinates of the point of contact.

 Ans : a 2 l 2 - b 2 m 2 = n 2 ,


 a2l
 ,

n

b 2 m  

n 

x2
y2
= 1 between the
b2
a2
point of contact and its intersection with a directrix subtends a right angle at the
corresponding focus.
( 20 ) Prove that the segment of the tangent to the hyperbola
-
( 21 ) Find the equations of the tangents drawn from the point ( - 2, - 1 ) to the hyperbola
2
2
2x - 3y = 6.
[ Ans: 3x - y + 5 = 0,
( 22 )
If the line
y = mx +
x - y + 1 = 0]
a 2m2 - b 2
point P ( α ), then prove that sin α =
( 23 )
If the lines
x
2
a
2
-
y
2
b
2
2
2y - x = 14
and
touches the hyperbola
2
a2
-
y2
b2
= 1
at the
b
.
am
3y - x = 9
= 1 , then find the values of a
[ Ans: a = 288, b = 33 ]
x2
2
are tangential to the hyperbola
2
and b .
PROBLEMS
06 - HYPERBOLA
Page 4
2
2
( 24 ) The tangent and normal at a point P on the rectangular hyperbola x - y = 1 cut off
intercepts a1, a2 on the X-axis and b1, b2 on the Y-axis. Prove that a1 a2 = b1 b2.
( 25 )
Prove that the locus of intersection of tangents to a hyperbola
x2
a2
-
y2
b2
= 1 , which
meet at a constant angle β, is the curve
2
2
2
2 2
(x + y + b - a )
( 26 )
2
2
2
= 4 cot β (a y
- b2x2 + a2 b2 ).
Prove that the equation of the chord of the hyperbola
mid-point at ( h, k ) is
hx
a2
-
ky
b2
h2
=
a2
-
k2
b2
x2
a2
-
y2
b2
= 1 which has its
.
( 27 )
If a rectangular hyperbola circumscribes a triangle, then prove that it also passes
through the orthocentre of the triangle.
( 28 )
If a circle and the rectangular hyperbola xy = c
“t3” and “t4”, then prove that
2
(i)
( ii )
meet in the four points “t1”, “t2”,
4
product of the abscissae of the four points = the product of their ordinates = c ,
the centre f the circle through the points “t1”, “t2”, “t3” is
 c 
 
 c 1
1
1
1

,
+
+
+ t 1 t 2 t 3  
  t 1 + t 2 + t 3 +


t1 t 2 t 3  2  t1
t2
t3
 2 
 
2
( 29 )
For a rectangular hyperbola xy = c , prove that the locus of the mid-points of the
2
2
2
2
chords of constant length 2d is ( x + y ) ( xy - c ) = d xy.
( 30 )
If P1, P2 and P3 are three points on the rectangular hyperbola xy = c , whose
abscissae are x1, x2 and x3, then prove that the area of the triangle P1 P2 P3 is
2
c2
2
( x1
- x 2 )( x 2 - x 3 )( x 3 - x 1 )
.
x1 x 2 x 3