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# 214 Objective Mathematics

(a) a
(c )b
2
(b) ab
(d ) b
3
8. Tangents are drawn from the points on the
line x - y - 5 = 0 to x + 4y
2
= 4, then all
the chords of contact pass through a fixed
point, whose coordinate are
(a)
(b)
Wl 7 - - T
(d) None of these
9. An ellipse has OB as semi-minor axis.
F and F' are its focii and the angle FBF' is
a right angle. Then eccentricity of the ellipse
is
(a) 1/ VT (b) 1/2
(c) 1/ Vf (d) None of these
10. The set of positive value of m for which a line
with slope m is a common tangent to ellipse
2 2
x y 2
r + ^r = 1 and parabola y = 4ax is given
a b
by
(a) (2, 0)
(c)(0, 1)
11. The eccentricity of the ellipse
ax
2
+ by
2
+ 2fx + 2gy + c = 0 if axis of
ellipse parallel to x-axis is
(b)(3, 5)
(d) None of these
(a)
W
(b)
(c)
+ b
+ b
None of these
12. The minimum length of the intercept of any
2 2
tangent on the ellipse
L. _
= 1 between
the coordinate axes is
(a) 2a (b) 2b
(c )a-b (d )a + b
13. If CP and CD are semi-conjugate diameters
2 2
of the ellipse ^ + ^ = 1, then CP
2
+ CD
2
=
a b
(a) a + b
(c) a
2
- b
1
(b) a
2
+b
2
(d) 4a
2
+ b
2
14. A man running round a race course notes that
the sum of the distances of two flag-posts
from him is always 10 metres and the
distance between the flag-posts is 8 metres.
The area of the path he encloses in square
metres is
(a) 15:t (b) 12n
(c) 187t (d) 8it
15. If the normal at the point P (<(>) to the ellipse
= 1 intersects it again at the point
14 5
Q (2<()), then cos <j) is equal to
2
3
3
(a)-
(c);
16. If a and P are eccentric angles of the ends of
2 2
x y
a focal chord of the ellipse z + ^
=
' > then
a P .
tan tan ^ is equal to
(a)
(c)
1 - e
1+e
e+1
e-1
(b)
(d)
e - 1
e+ 1
e-1
e + 3
17. The eccertricity of an ellipse
2
= 1
a"
whose latusrectum is half of its minor axis is
(a)
1
12
( c ) T
(b)

(d) None of these
18. The distances from the foci of P (a , b) on the
2 2
ellipse + = 1 are
( a ) 4 | b
( b ) 5 j a
Ellipse 215
( c) 5 ~b
(d) None of these
19. If the normal at an end of a latus rectum of
2 2
x y
the ellipse
2
+
2
= 1 passes through one
a b
extremity of the minor axis, then the
eccentricity of the ellipse is given by
(a) e
4
+ e
1
- 1 = 0
(b) e
2
+ e 4 = 0
(c)e = <2
(d) e = 3je
20. If A and B are two fixed points and P is a
variable point such that PA + PB = 4, the
locus of P is
(a) a parabola
(b) an ellipse
(c) a hyperbola
(d) None of these
21. The area of the parallelogram formed by the
tangents at the ends of conjugate diameters of
an ellipse is
(a) constant and is equal to the product of the
axes
(b) can not be constant
(c) constant and is equal to the two lines of the
product of the axes
(d) None of these
22. If C be the centre of the ellipse
2 2
9x + 16v = 144 and S is one focus, the ratio
of CS to major axis is
(a) VT: 16_ (b) : 4
(c) <5 : VT (d) None of these
23. The centre of the ellipse
(JC + y - 2 )
2
(x - Y )
2
, .
(a) (0, 0)
(c) (0, I)
(b)(1, 1)
(d) (1, 0)
24. The radius of the circle passing through the
2 2
JC y
foci of the ellipse T7 + "ir = 1. and having its
16 9
centre (0, 3) is
(a) 4
(c) V12
(b) 3
(d) 7/ 2
25. The length of the latus rectum of an ellipse is
one third of the major axis, its eccentricity
would be
(a) 2/3 (b) 1/V3"
(c) 1/V2 (d) <273
26. If the length of the major axis of an ellipse is
three times the length of its minor axis, its
eccentricity is
(a) 1/3 (b) 1/V3~
(c) 1/V2" (d) 2/ V2/ 3
27. An ellipse is described by using an endless
string which is passed over two pins. If the
axes are 6 cm and 4 cm, the necessary length
of the string and the distance between the
pins respectively in cms. are
(a) 6, 2 <5
(b) 6, V5
(c) 4, 2 VJ
(d) None of these
28. The locus of mid-points of a focal chord of
2 2
JC y
the ellipse + = 1 is
a b
2 2
, . x v ex
( a ) + =
a b
2
a
> 2
x ex
(b) - ^ =
a b
2
a
/ V 2 2 2 , 2
(c).v +y =a + b
(d) None of these
29. The locus of the point of intersection of
2 2
tangents to the ellipse + which
a' b~
meet at right angles, is
(a) a circle
(b) a parabola
(c) an ellipse
(d) a hyperbola
30. The number of real tangents that can be
2 1
drawn to the ellipse 3.v + 5y" = 32 passing
through (3, 5) is
(a) 0 ( b) l
(c)2 (d)4
31. Equation to the ellipse whose centre is (-2. 3)
and whose semi-axes are 3 and 2 and major
axis is parallel to the .v-axis, is given by
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