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JEE • MATHEMATICS • 2D GEOMETRY SYSTEM

2d geometry system practice · Easy

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 61-90 of 128 questions

61 Easy

61 of 128

Application +4 | -1

If the circumcentre of a triangle lies at the point

(a, a)

and the centroid is the mid-point of the line joining the points

(2 a+3, a+4)

and

(a-4,2 a-3)

; then the orthocentre of the triangle lies on the line

View answer and explanation

Correct option: D

(a+1) x-(a-1) y=2 a

Option D (

(a+1) x-(a-1) y=2 a

) is correct because the orthocenter lies on the line that satisfies the relation between circumcenter, centroid, and orthocenter.

62 Easy

62 of 128

Application +4 | -1

The locus of the point

(\sqrt{(3 h+2)}, \sqrt{3 k})

. If

(h, k)

lies on

x+y=1

View answer and explanation

Correct option: B

A circle

Option B (A circle) is correct because substituting

h = \frac{{x_1^2 - 2}}{{3}}

and

k = \frac{{y_1^2}}{{3}}

into

h + k = 1

results in the equation of a circle:

x^2 + y^2 = 5

63 Easy

63 of 128

Application +4 | -1

The equation of the locus of a point whose distance from

(a, 0)

is equal to its distance from

y

-axis, is

View answer and explanation

Correct option: B

y^{2}-2 a x+a^{2}=0

Option B (

y^{2}-2 a x+a^{2}=0

) is correct because the distance from the point

(x,y)

(a,0)

being equal to its distance from the y-axis (

x=0

) leads to the equation

(x-a)^2 + y^2 = x^2

, which simplifies to

y^{2}-2 a x+a^{2}=0

64 Easy

64 of 128

Application +4 | -1

P(1,2), Q(4,6), R(5,7)

and

S(a, b)

are the vertices of a parallelogram

P Q R S

, then

View answer and explanation

Correct option: C

a=2, b=3

Option C (a=2, b=3) is correct because in a parallelogram, the midpoints of the diagonals coincide, allowing us to solve for a and b.

65 Easy

65 of 128

Application +4 | -1

If two vertices of a triangle are

(6,4),(2,6)

and its centroid is

(4,6)

, then the third vertex is

View answer and explanation

Correct option: A

(4,8)

Option C (

a^{2}\left(m_{2}-m_{3}\right)\left(m_{3}-m_{1}\right)\left(m_{1}-m_{2}\right)

) is correct because it is the result of calculating the area of the triangle using the determinant formula.

66 Easy

66 of 128

Application +4 | -1

If the points

(-5,1),(p, 5)

and

(10,7)

are collinear, then the value of

p

will be

View answer and explanation

Correct option: A

Option A (5) is correct because substituting p=5 into the determinant equation for collinearity makes the determinant equal to zero.

67 Easy

67 of 128

Application +4 | -1

\alpha, \beta, \gamma

are the real roots of the equation

x^3-3 a x^2+3 b x-1=0

, then the centroid of the triangle with vertices

\left(\alpha, \frac{1}{\alpha}\right),\left(\beta, \frac{1}{\beta}\right)

and

\left(\gamma, \frac{1}{\gamma}\right)

is at the point

View answer and explanation

Correct option: A

(a, b)

Option A

(a, b)

is correct because the centroid's coordinates are derived directly from Vieta's formulas and the given equation's coefficients.

68 Easy

68 of 128

Application +4 | -1

The triangle with vertices

A(2,7), B(4, y)

and

C(-2,6)

is right angled at

A

View answer and explanation

Correct option: A

y=-1

Option A (

y=-1

) is correct because the product of the slopes of

AB

and

AC

must be -1 for the triangle to be right-angled at

A

69 Easy

69 of 128

Application +4 | -1

The triangle joining the points

P(-2,7), Q(-4,-1)

and

R(2,6)

View answer and explanation

Correct option: B

a right angled triangle

Option B (a right angled triangle) is correct because the square of one side equals the sum of the squares of the other two sides, satisfying the Pythagorean theorem.

70 Easy

70 of 128

Application +4 | -1

If the area of the triangle formed by the points

O(0,0), A\left(a^{x^2}, 0\right)

and

B\left(0, a^{6 x}\right)

\frac{1}{2 a^5}

sq. units, then

x=

View answer and explanation

Correct option: D

-1,-5

Option D (-1, -5) is correct because by equating the area of the triangle to the given value and solving the resulting exponential equation, we find x = -1 or x = -5.

71 Easy

71 of 128

Application +4 | -1

Centroid of the triangle, whose vertices are

(0,0),(2,5)

and

(7,4)

, is

View answer and explanation

Correct option: A

(3,3)

Option A (3,3) is correct because the centroid is calculated by averaging the x and y coordinates of the vertices.

72 Easy

72 of 128

Application +4 | -1

In a right angled triangle

A B C

right angled at

C

C A=a, C B=b

. If the angular points

A

and

B

slide along

x

-axis and

y

-axis respectively then

C

lies on

View answer and explanation

Correct option: B

a x \pm b y=0

Option A (

b x \pm a y=0

) is correct because it represents the locus of point C as A and B slide along the axes, maintaining the right angle at C.

73 Easy

73 of 128

Application +4 | -1

A point moves in such a way that the sum of square of its distance from the points

A(2,0)

and

B(-2,0)

is always equal to the square of the distance between

A

and

B

. The locus of the point is

View answer and explanation

Correct option: D

x^{2}+y^{2}-4=0

Option D (

x^{2}+y^{2}-4=0

) is correct because the sum of the squares of the distances from

(x,y)

(2,0)

and

(-2,0)

being equal to the square of the distance between

(2,0)

and

(-2,0)

leads to the equation

(x-2)^2 + y^2 + (x+2)^2 + y^2 = 16

, which simplifies to

x^{2}+y^{2}=4

x^{2}+y^{2}-4=0

74 Easy

74 of 128

Understanding +4 | -1

Point

P\left(5,60 ^ {\circ}\right)

having polar coordinates is equivalent to

View answer and explanation

Correct option: C

\left(5,420^{\circ}\right)

Option C ((5,420°)) is correct because adding multiples of 360° to the angle in polar coordinates results in an equivalent point.

75 Easy

75 of 128

Application +4 | -1

If the vertices of a triangle be

(a, 1),(b, 3)

and

(4, c)

, then the centroid of the triangle will lie on

x

-axis, if

View answer and explanation

Correct option: C

c=-4

Option C (

c=-4

) is correct because the centroid of the triangle lies on the x-axis, implying the y-coordinate of the centroid is zero, leading to

c=-4

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76 Easy

76 of 128

Application +4 | -1

The area of a triangle whose vertices are

(1,-1),(-1,1)

and

(-1,-1)

is given by

View answer and explanation

Correct option: A

Option A (2) is correct because it is the area of the triangle calculated using the determinant formula with the given vertices.

77 Easy

77 of 128

Application +4 | -1

The quadrilateral formed by the vertices

(-1,1),(0,-3),(5,2)

and

(4,6)

will be

View answer and explanation

Correct option: B

Parallelogram

Option B (Parallelogram) is correct. The quadrilateral is a parallelogram because opposite sides are equal in length, but the diagonals are not equal, indicating it is not a rectangle or square.

78 Easy

78 of 128

Application +4 | -1

If polar coordinates of any point are

\left(2, \frac{\pi}{3}\right)

, then its cartesian coordinates are

View answer and explanation

Correct option: B

(1, \sqrt{3})

Option B (

(1, \sqrt{{3}})

) is correct because it represents the Cartesian coordinates corresponding to the given polar coordinates.

79 Easy

79 of 128

Application +4 | -1

The quadrilateral

A B C D

formed by the points

A(0,0) ; B(3,4), C(7,7)

and

D(4,3)

is a

View answer and explanation

Correct option: C

rhombus

Option D (parallelogram) is correct because opposite sides are parallel, but not all sides are equal, and the angles are not necessarily right angles.

80 Easy

80 of 128

Application +4 | -1

Area of the triangle formed by the lines

y-x=0

x+y=0

and

y=k

in square units is

View answer and explanation

Correct option: B

k^2

Option D (

k^2 / 2

) is correct because the area of the triangle formed by the given lines is calculated using the coordinates of the vertices.

81 Easy

81 of 128

Application +4 | -1

Area of pentagon formed by

(1,2),(2,-1),(-2,-1),(2,1)

and

(-1,2)

View answer and explanation

Correct option: D

Option D, 10, is the correct area of the pentagon, calculated using the Shoelace formula.

82 Easy

82 of 128

Application +4 | -1

If the coordinates of the mid-points of side

A B

and

A C

\triangle A B C

are

D(3,5)

and

E(-3,-3)

respectively, the

B C=

View answer and explanation

Correct option: C

Option C (20) is correct because the length of BC is twice the length of the line segment joining the midpoints of AB and AC.

83 Easy

83 of 128

Application +4 | -1

A B C D

is a rectangle in the clockwise direction. The coordinates of

A

are

(1,3)

and of

C

are

(5,1)

, vertices

B

and

D

lie on the line

y=2 x+c

, then the coordinates of

D

are

View answer and explanation

Correct option: A

(2,0)

Option A (2,0) is correct because the midpoint of AC lies on the line y=2x+c, and AD is perpendicular to CD. Since D lies below A, its coordinates are (2,0).

84 Easy

84 of 128

Application +4 | -1

(x+3,4-y)=(1,7)

, then the value of

4+y

View answer and explanation

Correct option: A

Option A (1) is correct because solving for y gives y = -3, and thus 4 + y = 1.

85 Easy

85 of 128

Application +4 | -1

O

is the origin and

P\left(x_1, y_1\right), Q\left(x_2, y_2\right)

are two points, then

O P

OQ

\cos \angle P O Q

View answer and explanation

Correct option: A

x_1 x_2+y_1 y_2

Option A (

x_1 x_2+y_1 y_2

) is correct because it represents the dot product of the vectors OP and OQ.

86 Easy

86 of 128

Application +4 | -1

Let a variable line of slope

\mathrm{m} > 0

passing through the point

(4,-9)

intersect the coordinate axes at the points

\mathrm{A}

and

\mathrm{B}

. the minimum value of the sum of the distances of

\mathrm{A}

and

\mathrm{B}

from the origin is

View answer and explanation

Correct option: A

Option A (25) is correct because it is the minimum value of the sum of the distances of A and B from the origin, found using the AM-GM inequality.

87 Easy

87 of 128

Application +4 | -1

If the coordinates of two opposite vertices of a square are

(6,-3)

and

(-2,3)

, then area of square is

View answer and explanation

Correct option: A

50 \mathrm{sq}

units

Option A (50 sq units) is correct because the area of a square is half the square of the diagonal, and the diagonal is calculated correctly.

88 Easy

88 of 128

Application +4 | -1

A(4,-3), B(3,-2)

and

C(2,8)

are the vertices of a triangle, then its centroid will be

View answer and explanation

Correct option: C

(3,1)

Option C (

(3,1)

) is correct because the centroid is found by averaging the x-coordinates and y-coordinates of the vertices.

89 Easy

89 of 128

Application +4 | -1

A triangle with vertices

(4,0),(-1,-1)

(3,5)

, is

View answer and explanation

Correct option: A

isosceles and right angled

Option A (isosceles and right angled) is correct because the triangle satisfies both the conditions for being isosceles and right-angled.

90 Easy

90 of 128

Application +4 | -1

The points of trisection of line segment joining the points

A(2,1)

and

B(5,3)

, are

View answer and explanation

Correct option: C

\left(3, \frac{5}{3}\right),\left(4, \frac{7}{3}\right)

Option C ((3,

\frac{{5}}{{3}}

), (4,

\frac{{7}}{{3}}

)) is correct because these are the coordinates that divide the line segment into three equal parts.

Revision notes and key takeaways

Review 2D GEOMETRY SYSTEM in your textbook or class notes, then use this set to check your understanding.

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