Page 1 of 2. Guess Paper_December 2017. B.A./B.Sc. ( General ) Ist Semester. Paper â I : Plane Geometry. Time Allowed
Sanjay Gupta, Dev Samaj College For Women, Ferozepur City Guess Paper_December 2017 B.A./B.Sc. ( General ) Ist Semester Paper – I : Plane Geometry Time Allowed : Three Hours ] [ Maximum Marks : 30 Note : - Attempt any Five questions in all, selecting at least two questions from each Section. SECTION I I.(a) Through what angle, the axes should be rotated to remove the mixed term in the equation
5 x 2 4 x y 5 y 2 3 x 4 y 5 0 . Also find the transformed equation. Find the equation of two straight lines passing through points ( x1 , y1 ) and perpendicular to the pair of
(b)
straight lines a x 2 2 h x y b y 2 = 0.
(3,3)
II.(a) If a pair of lines x 2 2 p x y y 2 = 0 bisect the angle between the pairs of lines x 2 2 q x y y 2 = 0,
show that the later pair also bisects the angle between the first pair. (b) Find the equation of the pair of lines joining the origin of co-ordinates of the points of intersection of the line
y mx c with the curve x 2 y 2 a 2 . Prove that they are perpendicular if 2c 2 a 2 (1 m2 ) . (3,3) III.(a)
Find the equation of circle described on the common chord of circles x2 y 2 6 x 4 y 12 = 0 and
x2 y 2 2 x 6 y 15 = 0 as diameter. (b)
Find the locus of the middle points of the chords of the circle x y 8 x 4 y 5 0 which subtends a right angle at the centre of the circle. (3,3) 2
IV. (a) Find the length of the common chord of the circles
2
x 2 y 2 4 and x 2 y 2 2 x 4 y 1 0 .
Verify that the common chord is at right angles to the line joining the centres of the circles. (b) The circle x 2 y 2 4 x 6 y 3 = 0 is one of the circles of a co-axial family having line
2 x – 4 y + 1 = 0 as radical axis. Find circles of the system that touch the line x + 3 y – 2 = 0. (3 , 3 )
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Sanjay Gupta, Dev Samaj College For Women, Ferozepur City SECTION II V.(a) ) In the parabola
y 2 4ax , Show that the locus of the middle point of the normal PG at P , where G
is on the axis, is a parabola. (b
Prove that the locus of the poles of chords of the parabola y 2 = 4 a x which subtend a constant angle at the vertex is the curve ( x 4 a ) = 4 ( y 2 4 a x) cot 2 . 2
(3,3)
VI.(a) Find the equations of the tangents drawn from the point (4, 1) to the ellipse x 2 2 y 2 = 6. Also find
the angle between the tangents. (b) Find the length of the semi-diameter conjugate to the diameter
y 3x of the ellipse 9x2 + 4y2 = 36. (3,3)
VII.(a) P is variable point on the hyperbola
the middle point of AP is
x
2
a
2
( 2 x a) 2 a2
y
2
b2
4 y2 b2
= 1 whose vertex A is (a, 0). Show that the locus of = 1.
(b) Find joint equation of asymptotes of the hyperbola 3 x 5 xy 2 y equation of the conjugate hyperbola to given hyperbola. 2
2
5 x 11y 16 0 . Also, find
VIII.(a) Show that the poles of all normal chords of the rectangular hyperbola xy c curve
(3,3) 2
lie on the
(x y ) 4xyc 0 2
2 2
(b) Identify the curve eccentricity.
2
+4
+
− 2 + 2 − 6 = 0.
Also, find its centre, length of axes and (3,3)
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