Manual 4 Jee Linalg 2d
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JEE Problems in Linear
Abstract—A collection of problems from JEE mains
papers related to 2D coordinate geometry are available
in this document. These problems should be solved using
1. Tangent and normal are drawn at
on the parabola
xT 0 0
0 1!x+16 0x=0 (2)
which intersect the axis of the parabola at
Aand Brespectively. If Cis the centre of
the circle through the ponts P A and B, ﬁnd
2. A circle passes through the points 2
If its centre lies on the line
−1 4x+3=0 (3)
ﬁnd its radius.
3. Two parabolas with a common vertex and
with axes along x-axis and y-axis, respectively,
intersect each other in the ﬁrst quadrant. If the
length of the latus rectum of each parabola is
3, ﬁnd the equation of the common tangent to
the two parabolas.
4. If the tangents drawn to the hyperbola
V= 1 0
intersect the coordinate axes at the distinct
points Aand B, ﬁnd the locus of the mid point
5. βis one of the angles between the normals to
V= 1 0
at the points
3 cos θ
√3 sin θ!, −3 sin θ
√3 cos θ!, θ ∈0,π
then ﬁnd 2 cot β
6. The sides of a rhombus ABC are parallel to the
If the diagonals of the rhombus intersect at
and the vertex A(diﬀerent) from the origin is
on the y-axis, then ﬁnd the ordinate of A.
7. Tangents drawn from the point −8
xT 0 0
0 1!x+−8 0x=0 (12)
touch the parabola at Pand Q. If Fis the focus
of the parabola, then ﬁnd the area of △PFQ.
8. A normal to the hyperbola
xT 4 0
meets the coordinate axes xand yat Aand
Brespectively. If the parallelogram OABP is
formed, ﬁnd the locus of P.
9. Find the locus of the point of intersection of
√2k kx−4√2=0 (15)
10. If a circle C,whose radius is 3, touches exter-
nally the circle
at the point 2
2!, then ﬁnd the length of the
intercept cut by this circle Con the x-axis.
11. Let Pbe the parabola
xT 1 0
0 0!x+0 4x=0 (17)
Given that the distance of Pfrom the centre of
is minimum. Find the equation of the tangent
to the parabola at P.
12. The length of the latus rectum of an ellipse
is 4 ad the distance between a focus and its
nearest vertex on the major axis is 3
2. Find its
13. A square, of each side 2, lies above the x-axis
and has one vertex at the origin. If one of the
sides passing through the origin makes an angle
30◦with the positive direction of the x-axis,
then ﬁnd the sum of the x-coordinates of the
vertices of the square.
14. A line drawn through the point
cuts the circle
at the points Aand B. Find PA.PB.
15. Find the eccentricity of an ellipse having centre
at the origin, axes along the coordinate axes
and passing through the points
16. m−1x+c=0 is the normal at a point on
xT 0 0
0 1!x−8 0x=0=0 (22)
whose focal distance is 8. Find |c|.
17. Find the locus of the point of intersection of
the straight lines
18. The common tangents to the parabola
xT 1 0
0 0!x− 0
intersect at the point P. Find the distance of P
from the origin.
19. Consider an ellipse, whose centre is at the
origin and its major axis is along the x-axis. If
its eccentricity is 3
5and the distance between its
foci is 6, then ﬁnd the area of the quadrilateral
inscribed in the ellipse, with the vertices as the
vertices of the ellipse.
20. Let kbe an integer such that the triangle with
has area 28. Find the orthocentre of this trian-
21. A hyperbola passes through the point
and has foci at ±2
0!. Find the equation of the
tangent to this hyperbola at P.
22. If an equlateral triangle, having centroid at the
origin, has a side along the line
then ﬁnd the area of this triangle.
23. Find the equation of the circle, which is the
mirror image of the circle
xTx−2 0x=0 (29)
in the line 1 1x=3.(30)
24. Find the product of the perpendiculars drawn
from the foci of the ellipse
xT 25 0
0 9!x=225 (31)
upon the tangent to it at the point
25. Find the equation of the normal to the hyper-
xT 9 0
drawn at the point
26. Two sides of a rhombus are along the lines
If its diagonals intersect at
ﬁnd its vertices.
27. Find the locus of the centres of those circles
which touch the circle
xTx−81 1x=4 (38)
and also touch the x-axis.
28. One of the diameters of the circle, given by
xTx+2−2 3x=12 =0 (39)
is a chord of a circle S, whose centre is at
Find the radius of S.
29. Let Pbe the point on the parabola
xT 0 0
0 1!x−8 0x=0 (41)
which is at a minimum distance from the centre
Cof the circle
xTx+0 12x=1 (42)
Find the equation of the circle passing through
Cand having its centre at P.
30. Find the eccentricity of the hyperbola whose
length of the latus rectum is equal to 8 and the
length of its conjugate axis is equal to half the
distance between its foci.
31. A variable line drawn through the intersection
of the lines
4 3x=12 (43)
3 4x=12 (44)
meets the coordinate axes at Aand B, then ﬁnd
the locus of the midpoint of AB.
32. The point 2
is translated parallel to the line
by 2 √3 units. If the new point Qlies in the
third quadrant, then ﬁnd the equation of the
line passing through Qand perpendicular to L.
33. A circle passes through
and touches the y-axis at
Which one of the following equations can
represent a diameter of this circle?
a) 4 5x=6
b) 2−3x+10 =0
c) 3 4x=3
d) 5 2x+4=0
34. Let aand brespectively be the semi-transverse
and semi-conjugate axes of a hyperbola whose
eccentricity satisﬁes the equation
is a focus and
5 0x=9 (51)
is the corresponding directrix of this hyperbola,
then ﬁnd a2−b2.
35. A straight line through the origin Omeets the
4 3x=10 (52)
8 6x+5=0 (53)
at Aand Brespectively. Find the ratio in which
36. Find the equation of the tangent to the circle,
at the point 1
whose centre is the point of intersection of the
2 1x=3 (55)
37. Pand Qare two distinct points on the parabola
xT 0 0
0 1!x−4 0x=0 (57)
with parameters tand t1respectively. If the
normal at Ppasses through Q, then ﬁnd the
minimum value of t2
38. A hyperbola whose transverse axis is along the
major axis of the conic
V= 3 0
and has vertices at the foci of this conic. If the
eccentricity of the hyperbola is 3
2, which of the
following points doesnot lie on it?
39. A tangent at a point on the ellipse
V= 3 0
meets the coordinate axes at Aand B. If Obe
the origin, ﬁnd the minimum area of △OAB.