Q.7. If the tangent at P of the hyperbola xy = c
^{2} meets the asymptotes ate L & M & C is the centre of hyperbola. Prove that PL = PM = CP.
Ans:
Let P(ct,
) be any point, then equation of tangent at P is
x + t
^{2}y = 2ct
It meets the asymptotes i.e. x = 0 & y = 0 at L, M respectively.
L = (2ct, 0)
& M = (0,
)
Clearly mid point of LM is
= (ct,
) = P
Therefore PL = PM =
................................................... (i)
& CP =
But ML =
ML = 2
ML = 2CP
CP =
................................................. (ii)
By (i) & (ii)
PL = PM = CP
Q.8. Find the equation of hyperbola whose asymptotes are 2x + y + 2 = 0 & x + y + 3 = 0 & which passes through (0, 1), also find the equation of conjugate Hyperbola.
Ans: We know that the combined equation of asymptotes & hyperbola differ by a constant.
The combined equation of assymptotes is
(2x + y + 2)(x + y + 3) = 0
2x
^{2} + y
^{2} + 3xy + 8x + 5y + 6 = 0
Then let the equation of hyperbola be
2x
^{2} + y
^{2} + 3xy + 8x + 5y + 6 +
= 0
As it passes through (0, 1) we get
0 + 1 + 0 + 0 + 5 + 6 +
= 0
=  12
Hence the equation of hyperbola is
2x
^{2} + y
^{2} + 3xy + 8x + 5y  6 = 0
We also know that
The equation of conjugate hyperbola = 2(combined equation of assymptotes)  (equation of hyperbola)
equation of conjugate hyperbola is
2(2x
^{2} + y
^{2} + 3xy + 8x + 5y + 6)  (2x
^{2} + y
^{2} + 3xy + 8x + 5y  6)
equation of conjugate hyperbola is
2x
^{2} + y
^{2} + 3xy + 8x + 5y + 18 = 0
Q.9. If the polars of (x
_{1}, y
_{1}) & (x
_{2}, y
_{2}) with respect to hyperbola
= 1 are at right angles, then show that
= 0.
Ans: Equation of polar of (x
_{1}, y
_{1}) & (x
_{2}, y
_{2}) with respect to hyperbola
= 1 are
= 1 &
= 1
Therefore the slopes are
&
Since the polars are at right angles
x
=  1
Q.10. Prove that product of perpendiculars from any point on the hyperbola
= 1 to its asymptotes is
.
Ans: The asymptotes are
y =
& y = 
Let the point be P (x
_{1}, y
_{1})
Perpendicular distance from y =
is
L
_{1} =
Perpendicular distance from y = 
is
L
_{2} =
L
_{1}L
_{2} =
But
= 1 (x
_{1}, y
_{1} lies on hyperbola)
L
_{1}L
_{2} =
L
_{1}L
_{2} =
MEDIUM
Q.1. Show that the locus of the centre of circle which touches two given circles externally is a hyperbola.
Ans: Let C
_{1}, C
_{2} be the centres of the two given circles & p & q be their radii. Let
be the radius of the circle touching them externally & c be its centre, then,
CC
_{1} = p + r
CC
_{2} = q +
We can see that
CC
_{1}  CC
_{2} = p  q = constant
Here we observe that the difference of distances of the centre from two fixed points is constant. Hence it satisfies the property of hyperbola the locus of centre is a hyperbola.
Q.12. Find the locus of mid points mid of the chord of x
^{2} + y
^{2} = 16 which are tangets to the hyperbola
= 1.
Ans: The equation of chord of circle with (h, k) as mid point is
T = S
_{1}
i.e. hx + ky  16 =h
^{2} + k
^{2}  16
hx + ky = h
^{2} + k
^{2} .................................................... (i)
If (i) is the tangent to hyperbola therefore it must be of the form
sec

tan
= 1
here a = 2 b = 3
sec

tan
= 1 .................................................. (ii)
Equating coefficients of (i) & (ii)
& tan
= 
Using sec
^{2}  tan
^{2} = 1 we get
4h
^{2}  9k
^{2} = (h
^{2} + k
^{2})
^{2}
Locus of (h, k) is (x
^{2} + y
^{2})
^{2} = 4x
^{2}  9y
^{2}
Q.13. A circle with cintre (3
, 2
) & of variable radius cuts the rectangular hyperbola x
^{2}  y
^{2} = 9a
^{2} at the points A, B, C, D. Find the locus of centroid of triangle ABC ?
Ans: The equation of circle is
(x  3
)
^{2} + (y  3
)
^{2} =
^{2} .................................. (i)
& hyperbola is
x
^{2}  y
^{2} = 9a
^{2} ..................................................... (ii)
Eliminating y from (i) & (ii) we get
4x
^{4}  24
x
^{3} + .............................. = 0
It is a equation of power four, having roots as x
_{1}, x
_{2}, x
_{3}, x
_{4}.
Let (h, k) be the centroid of PQR
Then,
h =
& k =
We have x
_{1} + x
_{2} + x
_{3} + x
_{4} = 6
................................. (iii)
& similarly y
_{1} + y
_{2} + y
_{3} + y
_{4} = 6