Download PTU (I.K. Gujral Punjab Technical University Jalandhar (IKGPTU) B.Sc Hons (Bachelor of Science Honours) 2020 March Previous Question Papers
1 | M-77313 (S1)-2386
Roll No. Total No. of Pages : 02
Total No. of Questions : 09
Bachelor of Science - Honours (Mathematics) (Sem.?1)
CO-ORDINATE GEOMETRY
Subject Code : UC-BSHM-102-19
M.Code : 77313
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION - B & C. have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.
SECTION-A
1. Solve the following :
a) For what values of k does the equation 12x
2
? 10xy + 2y
2
+ 11x ? 5y + k = 0 represent
two straight lines?
b) Find the equations of the tangent and normal at the point of the parabola y
2
= 8x
whose ordinate is 4.
c) Find the pole of the line x ? 2y + 3 = 0 w.r.t. the ellipse 3x
2
+ 4y
2
= 12.
d) If e
1
and e
2
be the eccentricities of a hyperbola and of the conjugate hyperbola, then
show that
2 2
1 2
1 1
1
e e
? ? .
e) Show that the equation r
2
? 2br cos ( ? ? ?)= c describes a circle with centre (b, ?) if
b
2
+ c > 0.
f) Show that if ax
2
+ 2hxy + by
2
= 1 and a
/
x
2
+ 2h
/
xy + b
/
y
2
= 1 represent the same
conic and the axes are rectangular, then (a ? b)
2
+ 4h
2
= (a
/
? b
/
)
2
+
2
/
4h .
g) Find the equation to a circle, the axis of coordinates being two straight lines through
its centre at right angles.
h) Find the equations of the tangents to the circle x
2
+ y
2
? 6x + 4y = 12 which are
parallel to the line 4x + 3y + 5 = 0.
i) Prove that the circles x
2
+ y
2
? 2ax + c = 0 and x
2
+ y
2
+ 2by ? c = 0 intersect
orthogonally.
j) If pairs of straight lines x
2
? 2pxy ? y
2
= 0 and x
2
? 2qxy ? y
2
= 0 be such that each pair
bisects the angle between the other pair, prove that pq = ?1.
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1 | M-77313 (S1)-2386
Roll No. Total No. of Pages : 02
Total No. of Questions : 09
Bachelor of Science - Honours (Mathematics) (Sem.?1)
CO-ORDINATE GEOMETRY
Subject Code : UC-BSHM-102-19
M.Code : 77313
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks
each.
2. SECTION - B & C. have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.
SECTION-A
1. Solve the following :
a) For what values of k does the equation 12x
2
? 10xy + 2y
2
+ 11x ? 5y + k = 0 represent
two straight lines?
b) Find the equations of the tangent and normal at the point of the parabola y
2
= 8x
whose ordinate is 4.
c) Find the pole of the line x ? 2y + 3 = 0 w.r.t. the ellipse 3x
2
+ 4y
2
= 12.
d) If e
1
and e
2
be the eccentricities of a hyperbola and of the conjugate hyperbola, then
show that
2 2
1 2
1 1
1
e e
? ? .
e) Show that the equation r
2
? 2br cos ( ? ? ?)= c describes a circle with centre (b, ?) if
b
2
+ c > 0.
f) Show that if ax
2
+ 2hxy + by
2
= 1 and a
/
x
2
+ 2h
/
xy + b
/
y
2
= 1 represent the same
conic and the axes are rectangular, then (a ? b)
2
+ 4h
2
= (a
/
? b
/
)
2
+
2
/
4h .
g) Find the equation to a circle, the axis of coordinates being two straight lines through
its centre at right angles.
h) Find the equations of the tangents to the circle x
2
+ y
2
? 6x + 4y = 12 which are
parallel to the line 4x + 3y + 5 = 0.
i) Prove that the circles x
2
+ y
2
? 2ax + c = 0 and x
2
+ y
2
+ 2by ? c = 0 intersect
orthogonally.
j) If pairs of straight lines x
2
? 2pxy ? y
2
= 0 and x
2
? 2qxy ? y
2
= 0 be such that each pair
bisects the angle between the other pair, prove that pq = ?1.
2 | M-77313 (S1)-2386
SECTION-B
2. Find the angle between the straight lines given by the equation ax
2
+ 2hxy + by
2
= 0.
Also find the condition of perpendicularity.
3. Find the equations of the lines joining the origin to the points of intersection of the line
2x ? 3y + 4 = 0 with the curve x
2
+ 4xy + 2y
2
+ 12x + 4y = 0, and show that they are at
right angles.
4. a) Find the pole of the straight line 9x + y ? 28 = 0 with respect to the circle
2x
2
+ 2y
2
? 3x + 5y ? 7 = 0.
b) Find the lengths of the tangents drawn to the circle 3x
2
+ 3y
2
? 7x ? 6y = 12 from the
point (6, ?7).
5. Find the locus of a point P which is such that its polar with respect to one circle touches a
second circle.
SECTION-C
6. a) Prove that the locus of the poles of the chords which are normal to the parabola
y
2
= 4ax is the curve y
2
(x +2a) + 4a
3
= 0.
b) Prove that the chord of a parabola which subtends a right angle at the vertex meets its
axis in a fixed point.
7. a) If the normal at the end of a latus rectum of an ellipse passes through one extremity of
the minor axis, show that the eccentricity of the curve is given by the equation
e
4
+ e
2
? 1 = 0.
b) The chord of contact of tangents from a point P to the hyperbola
2 2
2 2
1
x y
a b
? ? subtends
a right angle at the centre. Prove that locus of P is the ellipse
2 2
4 4 2 2
1 1 x y
a b a b
? ? ? .
8. a) Through what angle should the axes be rotated so that the mixed term may disappear
from the equation 17x
2
? 16xy + 17y
2
? 225 = 0?
b) On shifting the origin to the point (1, ?1), the axis remaining parallel to the original
axis, the equation of a curve becomes 4x
2
+ y
2
+ 3x ? 4y + 2 = 0. Find its original
equation.
9. a) Identify the curve represented by the equation 3x
2
+ 2xy + 3y
2
+ 18x + 22y + 50 = 0.
Reduce it to standard form by suitable transformation of axes.
b) Find the equation of the chord of contact of tangents drawn from the point (r
1
, ?
1
) to
the conic 1 cos
l
e
r
? ? ?.
NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any
page of Answer Sheet will lead to UMC against the Student.
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This post was last modified on 02 April 2020