# Manual 4 Jee Linalg 2d

### Manual%204%20jee_linalg_2d

### Manual%204%20jee_linalg_2d

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JEE Problems in Linear

Algebra: 2D

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

linear algebra.

1. Tangent and normal are drawn at

P= 16

16!(1)

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

tan CPB.

2. A circle passes through the points 2

3!and 4

5!.

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

xTVx+1=0 (4)

where

V= 1 0

0−4!(5)

intersect the coordinate axes at the distinct

points Aand B, ﬁnd the locus of the mid point

of AB.

5. βis one of the angles between the normals to

the ellipse

xTVx=9 (6)

where

V= 1 0

0 3!(7)

at the points

3 cos θ

√3 sin θ!, −3 sin θ

√3 cos θ!, θ ∈0,π

2,(8)

then ﬁnd 2 cot β

sin 2θ.

6. The sides of a rhombus ABC are parallel to the

lines

1−1x+2=0 (9)

7−1x+3=0.(10)

If the diagonals of the rhombus intersect at

P= 1

2!(11)

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

0!to the

parabola

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

0−9!x=36 (13)

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

the lines

√2−1x+4√2k=0 (14)

√2k kx−4√2=0 (15)

10. If a circle C,whose radius is 3, touches exter-

2

nally the circle

xTx+2−4x=4 (16)

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

the circle

xTx+ 6

0!x+8=0 (18)

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

eccentricity.

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

P= 4

7!(19)

cuts the circle

xTx=9 (20)

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

P= 4

−1!,Q= −2

2!.(21)

16. m−1x+c=0 is the normal at a point on

the parabola

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

t−2x−3t=0 (23)

1−2tx+3=0 (24)

18. The common tangents to the parabola

xT 1 0

0 0!x− 0

4!x=0=0 (25)

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

vertices k

−3k!, 5

k!, −k

2!(26)

has area 28. Find the orthocentre of this trian-

gle.

21. A hyperbola passes through the point

P= √2

√3!(27)

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

1 1x=2,(28)

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)

3

upon the tangent to it at the point

1

2 3

5√3!(32)

25. Find the equation of the normal to the hyper-

bola

xT 9 0

0−16!x=144 (33)

drawn at the point

8

3√3!(34)

26. Two sides of a rhombus are along the lines

1−1x+1=0 (35)

7−1x−5=0.(36)

If its diagonals intersect at

−1

−2!,(37)

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

−3

2!.(40)

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

1!(45)

is translated parallel to the line

1−1x=4 (46)

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

−2

4!(47)

and touches the y-axis at

0

2!.(48)

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

9e2−18e+5=0 (49)

If

S= 5

0!(50)

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

lines

4 3x=10 (52)

8 6x+5=0 (53)

at Aand Brespectively. Find the ratio in which

Odivides AB.

36. Find the equation of the tangent to the circle,

at the point 1

−1!,(54)

whose centre is the point of intersection of the

straight lines

2 1x=3 (55)

1−1x=1 (56)

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

1.

38. A hyperbola whose transverse axis is along the

major axis of the conic

xTVx=51 (58)

where

V= 3 0

0 27!(59)

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?

a) 0

2!

b) √5

2√2!

c) √10

2√3!

d) 5

2√3!

39. A tangent at a point on the ellipse

xTVx=51 (60)

where

V= 3 0

0 27!(61)

meets the coordinate axes at Aand B. If Obe

the origin, ﬁnd the minimum area of △OAB.