Values of $\alpha$ if $(\alpha, 2 \alpha)$ lies inside the $\Delta A B C$ if $A(0,2), B(2,0)$ and $C(4,4)$

$\alpha \in\left(\frac{1}{3}, \frac{2}{3}\right)$

$\alpha \in\left(\frac{2}{3}, 1\right)$

$\alpha \in\left(\frac{1}{3}, 1\right)$

Back to 2D GEOMETRY SYSTEM

Your Progress

0 / 187 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • 2D GEOMETRY SYSTEM

2d geometry system practice · Mixed difficulty

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

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.

Take a chapter quiz →

Showing 1-30 of 187 questions

1 Medium

1 of 187

Application +4 | -1

Values of

\alpha

(\alpha, 2 \alpha)

lies inside the

\Delta A B C

A(0,2), B(2,0)

and

C(4,4)

View answer and explanation

Correct option: C

\alpha \in\left(\frac{2}{3}, \frac{4}{3}\right)

Option C (

\alpha \in\left(\frac{{2}}{{3}}, \frac{{4}}{{3}}\right)

) is correct because the point

(\alpha, 2\alpha)

must satisfy the inequalities derived from the equations of the lines forming the triangle.

2 Medium

2 of 187

Application +4 | -1

The locus of the moving point

P

, such that

2 P A=3 P B

where

A

(0,0)

and

B

(4,-3)

, is

View answer and explanation

Correct option: D

5 x^{2}+5 y^{2}-72 x+54 y+225=0

Option D (

5 x^{2}+5 y^{2}-72 x+54 y+225=0

) is correct because it represents the locus of point P satisfying the given distance condition.

3 Easy

3 of 187

Application +4 | -1

The incentre of the triangle with vertices

(1, \sqrt{3}),(0,0)

and

(2,0)

View answer and explanation

Correct option: D

(1,1 / \sqrt{3})

Option D (1, 1/

\sqrt{3}

) is correct because the triangle is equilateral, so the incenter coincides with the centroid, which is easily calculated.

4 Easy

4 of 187

Application +4 | -1

The triangle with vertex

(1,5),(-3,1) \,\&\,(-2,1)

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the triangle's side lengths do not satisfy the conditions for being isosceles, equilateral, or a right-angled triangle.

5 Easy

5 of 187

Application +4 | -1

The area of the quadrilateral with its vertices at the foci of the conics

9 x^2-16 y^2-18 x+32 y-23

=0

and

25 x^2+9 y^2-50 x-18 y+33=0

, is

View answer and explanation

Correct option: B

\frac{8}{9}

Option B

\frac{8}{9}

is correct because calculating the foci of both conics and then finding the area of the quadrilateral formed by them gives this result.

6 Easy

6 of 187

Application +4 | -1

Orthocentre of the triangle whose vertices are

(0,0)(3,0)

and

(0,4)

View answer and explanation

Correct option: A

(0,0)

Option A (

(0,0)

) is correct because the orthocenter of a right-angled triangle is the vertex at the right angle, which is the origin in this case.

7 Easy

7 of 187

Analyzing +4 | -1

A triangle

A B C

with vertices

A(-1,0), B\left(-2, \frac{3}{4}\right)

and

C\left(-3,-\frac{7}{6}\right)

has its orthocentre at

H

. Then, the orthocentre of triangle

B C H

will be

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the orthocenter of triangle BCH is actually vertex A(-1, 0).

8 Easy

8 of 187

Application +4 | -1

If the vertices of a triangle be

(a, b-c),(b, c-a)

and

(c, a-b)

, then the centroid of the triangle lies

View answer and explanation

Correct option: B

x

-axis

Option B (On

x

-axis) is correct because the y-coordinate of the centroid is 0, indicating that it lies on the x-axis.

9 Easy

9 of 187

Application +4 | -1

The greatest angle of a cyclic quadrilateral is 3 times the least. The circular measure of the least angle is

View answer and explanation

Correct option: B

\frac{\pi}{4}

Option B (

\frac{\pi}{4}

) is correct because in a cyclic quadrilateral, opposite angles are supplementary. If the greatest angle is 3 times the least, then the least angle is 45 degrees, which is

\frac{\pi}{4}

radians.

10 Medium

10 of 187

Application +4 | -1

The ratio in which the line segment joining the points

(3,-4)

and

(-5,6)

is divided by the

X

-axis, is

View answer and explanation

Correct option: A

2: 3

Option A (2:3) is correct because the X-axis divides the line segment in the ratio 2:3, which is found by setting the y-coordinate of the section formula to zero.

11 Medium

11 of 187

Application +4 | -1

Locus of the points which are at equal distance from

3 x+4 y-11=0

and

12 x+5 y+2=0

and which is near the origin is

View answer and explanation

Correct option: B

99 x+77 y-133=0

Option B (

99 x+77 y-133=0

) is correct because it represents the locus of points equidistant from the two given lines and near the origin.

12 Easy

12 of 187

Application +4 | -1

The distance between points

P(-3,-2)

and

Q(-6,7)

, the axes being inclined at

60^{\circ}

, is

View answer and explanation

Correct option: A

3 \sqrt{7}

Option A (

3 \sqrt{{7}}

) is correct because it represents the distance between the points when the axes are inclined at 60 degrees.

13 Medium

13 of 187

Application +4 | -1

If the coordinates of a point be given by the equation

x=a(1-\cos \theta), y=a \sin \theta

, then the locus of the point will be

View answer and explanation

Correct option: B

A circle

Option B (A circle) is correct because eliminating

\theta

from the given parametric equations results in the equation

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

, which represents a circle.

14 Easy

14 of 187

Application +4 | -1

The points

(k-1, k+2),(k, k+1),(k+1, k)

are collinear for

View answer and explanation

Correct option: A

any value of

k

Option A (any value of

k

) is correct because the slope between any two points is independent of k, indicating collinearity for all k.

15 Easy

15 of 187

Application +4 | -1

A B C D

is a quadrilateral.

P(3,7)

and

Q(7,3)

are the middle points of the diagonals

A C

and

B D

respectively. The coordinates of the mean point (or the centre of mean position) of the vertices of the quadrilateral are

View answer and explanation

Correct option: C

(5,5)

Option C (5,5) is correct because the coordinates of the mean point are the midpoint of the line segment joining the midpoints of the diagonals.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

16 Medium

16 of 187

Understanding +4 | -1

A point moves in such a way that its distance from origin is always 4 . Then the locus of the point is

View answer and explanation

Correct option: B

x^{2}+y^{2}=16

Option B (

x^{2}+y^{2}=16

) is correct because the distance from the origin is 4, which translates to the equation of a circle centered at the origin with radius 4.

17 Easy

17 of 187

Application +4 | -1

(4 x+3, y)=(3 x+5,-2)

, then the sum of the values of

x

{y}

View answer and explanation

Correct option: A

Option A (0) is correct because solving the equations yields x = 2 and y = -2, so x + y = 0.

18 Easy

18 of 187

Application +4 | -1

One of the diameter of the circle circum scribing the rectangle

\mathrm{ABCD}

4 y=x+7

. If

\mathrm{A}

and B are points

(-3,4) \&(5,4)

respectively. Then the area of the rectangle is (In Sq units)

View answer and explanation

Correct option: D

Option D (32) is correct. By finding the center of the circumscribing circle and using the given points, the dimensions of the rectangle can be determined, leading to the area.

19 Easy

19 of 187

Application +4 | -1

P

and

Q

are two points whose coordinates are

\left(a t^2, 2 a t\right)

and

\left(\frac{a}{t^2},-\frac{2 a}{t}\right)

respectively and

S

is the point

(a, 0)

. Then, harmonic mean of

S P

and

S Q

, is

View answer and explanation

Correct option: C

2 a

Option C (

2a

) is correct because it represents the harmonic mean of the distances SP and SQ.

20 Easy

20 of 187

Understanding +4 | -1

If the point

\left(x_1+t\left(x_2-x_1\right), y_1+t\left(y_2-y_1\right)\right)

divides the join of

\left(x_1, y_1\right)

and

\left(x_2, y_2\right)

internally, then

View answer and explanation

Correct option: B

0 < t < 1

Option B (

0 < t < 1

) is correct because for internal division, the ratio must be positive and finite, implying t lies between 0 and 1.

21 Easy

21 of 187

Application +4 | -1

The incentre of the triangle with vertices

(1, \sqrt{3}),(0,0)

and

(2,0)

View answer and explanation

Correct option: D

\left(1, \frac{1}{\sqrt{3}}\right)

Option D

\left(1, \frac{1}{\sqrt{3}}\right)

is correct because the triangle is equilateral, making the incenter the same as the centroid, which is easily calculated.

22 Easy

22 of 187

Application +4 | -1

The mid points of the sides

A B

and

A C

of a triangle

A B C

are

(2,-1)

and

(-4,7)

respectively, then the length of

B C

View answer and explanation

Correct option: B

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

23 Easy

23 of 187

Application +4 | -1

If the distance between the points

(x, 2)

and

(3,4)

is 2 , then the value of

x

View answer and explanation

Correct option: C

Option C (3) is correct because it is the value of

x

that satisfies the given distance condition.

24 Medium

24 of 187

Application +4 | -1

Without changing the direction of coordinate axes, origin is transferred to

(h, k)

, so that the linear (one degree) terms in the equation

x^{2}+y^{2}-4 x+6 y-7=0

are eliminated. Then the point

(h, k)

View answer and explanation

Correct option: C

(2,-3)

Option C (

(2,-3)

) is correct because substituting

x=x'+h

and

y=y'+k

into the equation and setting the coefficients of

x'

and

y'

to zero eliminates the linear terms, leading to

h=2

and

k=-3

25 Medium

25 of 187

Application +4 | -1

If the coordinates of a point be given by the equations

x=b \sec \phi, \quad y=a \tan \phi

, then its locus is

View answer and explanation

Correct option: D

A hyperbola

Option D (A hyperbola) is correct because the given parametric equations,

x = b \sec \phi

and

y = a \tan \phi

, satisfy the equation of a hyperbola.

26 Hard

26 of 187

Application +4 | -1

A point moves such that the sum of its distances from two fixed points

(a e, 0)

and

(-a e, 0)

is always

2 a

. Then equation of its locus is

View answer and explanation

Correct option: A

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}\left(1-e^{2}\right)}=1

Option A (

\frac{{x^{2}}}{{a^{2}}}+\frac{{y^{2}}}{{a^{2}\left(1-e^{2}\right)}}=1

) is correct because it represents the equation of an ellipse where the sum of distances from two foci is constant.

27 Easy

27 of 187

Application +4 | -1

Let the area of the triangle with vertices

A(1,\alpha ),\,\,\,\,B(\alpha ,0)\,\,\,{\text{and}}\,\,\,\,\,C(0,\alpha )

be 4 sq units. If the points

(\alpha , - \alpha ),\,\,\,( - \alpha ,\alpha )\,\,\,\,{\text{and}}\,\,\,({\alpha ^2},\beta )

are collinear, then

\beta

is equal to

View answer and explanation

Correct option: C

-64

Option C (-64) is correct because by calculating the area of the triangle and using the collinearity condition, we can find the value of

\beta

28 Easy

28 of 187

Application +4 | -1

O

is the origin and

A

is the point

(3,4)

. If a point

P

moves so that the line segment

O P

is always parallel to the line segment

O A

, then the equation to the locus of

P

View answer and explanation

Correct option: A

4 x-3 y=0

Option A (

4x - 3y = 0

) is correct because the locus of point P, where OP is parallel to OA, results in a linear equation representing a straight line through the origin with the same slope as OA.

29 Easy

29 of 187

Understanding +4 | -1

The points

(a, b),(c, d)

and

\left(\frac{k c+l a}{k+l}, \frac{k d+l b}{k+l}\right)

are

View answer and explanation

Correct option: D

Collinear

Option D (Collinear) is correct because the third point is a section formula representation, indicating it lies on the line joining the first two points.

30 Medium

30 of 187

Application +4 | -1

In an equilateral triangle of side 

2\sqrt 3

cm, the circum radius is

View answer and explanation

Correct option: C

2 cm

Option C (2 cm) is correct because the circumradius R of an equilateral triangle is given by

R = \frac{a}{{2\sin A}}

, where a is the side length and A is the angle, which simplifies to 2 cm in this case.

Revision notes and key takeaways

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

Key Takeaways

✓ This set contains 187 single-correct-answer MCQs on 2D GEOMETRY SYSTEM.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

Showing 1-30 of 187 questions

Continue Your Practice

📚 Switch Set (ALL)

Chapter Sets

ALL 187 MCQs EASY 128 MCQs HARD 2 MCQs MEDIUM 57 MCQs

0 /187

0✓ 0✗

0% Accuracy