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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