# JEE Main

Explore popular questions from Coordinate Geometry for JEE Main. This collection covers Coordinate Geometry previous year JEE Main questions hand picked by experienced teachers.

## Mathematics

Coordinate Geometry

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Q 1. For every point {tex} P ( x , y , z ) {/tex} on the {tex} x y {/tex} -plane,

A

{tex} x = 0 {/tex}

B

{tex} y = 0 {/tex}

{tex} z = 0 {/tex}

D

None of these

##### Explanation

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Q 2. For every point {tex} P ( x , y , z ) {/tex} on the {tex} x {/tex} -axis (except the origin)

A

{tex} x = 0 , y = 0 , z \neq 0 {/tex}

B

{tex} x = 0 , z = 0 , y \neq 0 {/tex}

{tex} y = 0 , z = 0 , x \neq 0 {/tex}

D

None of these

##### Explanation

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Q 3. The distance of the point {tex} P ( a , b , c ) {/tex} from the {tex} x {/tex} -axis is

{tex} \sqrt { b ^ { 2 } + c ^ { 2 } } {/tex}

B

{tex} \sqrt { a ^ { 2 } + c ^ { 2 } } {/tex}

C

{tex} \sqrt { a ^ { 2 } + b ^ { 2 } } {/tex}

D

None of these

##### Explanation

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Q 4. Point {tex} ( - 3,1,2 ) {/tex} lies in

A

Octant I

Octant II

C

Octant III

D

Octant IV

##### Explanation

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Q 5. The three vertices of a parallelogram taken in order are {tex} ( - 1,0 ) , ( 3,1 ) {/tex} and {tex} ( 2,2 ) {/tex} respectively. The coordinate of the fourth vertex is

A

{tex} ( 2,1 ) {/tex}

{tex} ( - 2,1 ) {/tex}

C

{tex} ( 1,2 ) {/tex}

D

{tex} ( 1 , - 2 ) {/tex}

##### Explanation

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Q 6. The point equidistant from the four points {tex} ( 0,0,0 ) , ( 3 / 2,0,0 ) , {/tex} {tex} ( 0,5 / 2,0 ) {/tex} and {tex} ( 0,0,7 / 2 ) {/tex} is:

A

{tex} \left( \frac { 2 } { 3 } , \frac { 1 } { 3 } , \frac { 2 } { 5 } \right) {/tex}

B

{tex} \left( 3,2 , \frac { 3 } { 5 } \right) {/tex}

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

D

{tex} \left( \frac { 1 } { 2 } , 0 , - 1 \right) {/tex}

##### Explanation

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Q 7. {tex} \mathrm { P } ( \mathrm { a } , \mathrm { b } , \mathrm { c } ) ; \mathrm { Q } ( \mathrm { a } + 2 , \mathrm { b } + 2 , \mathrm { c } - 2 ) {/tex} and {tex} \mathrm { R } ( \mathrm { a } + 6 , \mathrm { b } + 6 , \mathrm { c } - 6 ) {/tex} are collinear.
Consider the following statements:
I. {tex} \mathrm { R}{/tex} divides {tex} \mathrm { PQ}{/tex} internally in the ratio 3: 2
II. {tex} \mathrm { R}{/tex} divides {tex} \mathrm { PQ}{/tex} externally in the ratio 3: 2
III. {tex} \mathrm { Q}{/tex} divides {tex} \mathrm { PR}{/tex} internally in the ratio 1: 2
Which of the statements given above is/are correct?

A

Only I

B

Only II

C

I and III

II and IIII

##### Explanation

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Q 8. The perpendicular distance of the point {tex} P ( 6,7,8 ) {/tex} from {tex} x y {/tex} -plane is

8

B

7

C

6

D

None of these

##### Explanation

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Q 9. The ratio in which the join of points {tex} ( 1 , - 2,3 ) {/tex} and {tex} ( 4,2 , - 1 ) {/tex} is divided by {tex} X O Y {/tex} plane is

A

{tex} 1:3{/tex}

{tex} 3:1{/tex}

C

{tex} -1 : 3{/tex}

D

None of these

##### Explanation

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Q 10. {tex} L {/tex} is the foot of the perpendicular drawn from a point {tex} P ( 6,7,8 ) {/tex} on the {tex} x y {/tex} -plane. The coordinates of point {tex} L {/tex} is

A

{tex} ( 6,0,0 ) {/tex}

{tex} ( 6,7,0 ) {/tex}

C

{tex} ( 6,0,8 ) {/tex}

D

None of these

##### Explanation

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Q 11. The equation of set points {tex} P {/tex} such that {tex} P A ^ { 2 } + P B ^ { 2 } = 2 K ^ { 2 } {/tex}, where {tex} A {/tex} and {tex} B {/tex} are the points {tex} ( 3,4,5 ) {/tex} and {tex} ( - 1,3 , - 7 ) , {/tex} respectively is

A

{tex} K ^ { 2 } - 109 {/tex}

{tex} 2 K ^ { 2 } - 109 {/tex}

C

{tex} 3 K ^ { 2 } - 109 {/tex}

D

{tex} 4 K ^ { 2 } - 10 {/tex}

##### Explanation

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Q 12. The co-ordinates of the point which divides the join of the points {tex} ( 2 , - 1,3 ) {/tex} and {tex} ( 4,3,1 ) {/tex} in the ratio {tex}3: 4{/tex} internally are given by:

A

{tex} \frac { 2 } { 7 } , \frac { 20 } { 7 } , \frac { 10 } { 7 } {/tex}

B

{tex} \frac { 10 } { 7 } , \frac { 15 } { 7 } , \frac { 2 } { 7 } {/tex}

{tex} \frac { 20 } { 7 } , \frac { 5 } { 7 } , \frac { 15 } { 7 } {/tex}

D

{tex} \frac { 15 } { 7 } , \frac { 20 } { 7 } , \frac { 3 } { 7 } {/tex}

##### Explanation

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Q 13. Let L, M, N be the feet of the perpendiculars drawn from a point P {tex} ( 7,9,4 ) {/tex} on the {tex} x , y {/tex} and {tex} z {/tex} -axes respectively. The coordinates of L, M and N respectively are

{tex} ( 7,0,0 ) , ( 0,9,0 ) , ( 0,0,4 ) {/tex}

B

{tex} ( 7,0,0 ) , ( 0,0,9 ) , ( 0,4,0 ) {/tex}

C

{tex} ( 0,7,0 ) , ( 0,0,9 ) , ( 4,0,0 ) {/tex}

D

{tex} ( 0,0,7 ) , ( 0,9,0 ) , ( 4,0,0 ) {/tex}

##### Explanation

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Q 14. If L, M and N are the feet of perpendiculars drawn from the point P(3, 4,5) on the XY, YZand ZX-planes respectively, then

A

distance of the point L from the point P is 5 units.

B

distance of the point M from the point P is 3 units.

C

distance of the point N from the point P is 4 units.

All of the above.

##### Explanation

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Q 15. Consider the following statements
I. The {tex} x{/tex}-axis and {tex} y {/tex}-axis together determine a plane known as {tex} xy {/tex}-plane.
II. Coordinates of points in {tex} xy {/tex}-plane are of the form {tex} \left( x _ { 1 } , y _ { 1 } , 0 \right) {/tex}
Choose the correct option.

A

Only I is true.

B

Only II is true.

Both are true

D

Both are false.

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Q 16. If the point {tex} A ( 3,2,2 ) {/tex} and {tex} B ( 5,5,4 ) {/tex} are equidistant from {tex} P {/tex}, which is on {tex} x {/tex} -axis, then the coordinates of {tex} P {/tex} are

A

{tex} \left( \frac { 39 } { 4 } , 2,0 \right) {/tex}

B

{tex} \left( \frac { 49 } { 4 } , 2,0 \right) {/tex}

C

{tex} \left( \frac { 39 } { 4 } , 0,0 \right) {/tex}

{tex} \left( \frac { 49 } { 4 } , 0,0 \right) {/tex}

##### Explanation

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Q 17. What is the locus of a point which is equidistant from the points {tex} ( 1,2,3 ) {/tex} and {tex} ( 3,2 , - 1 ) ? {/tex}

A

{tex} x + z = 0 {/tex}

B

{tex} x - 3 z = 0 {/tex}

C

{tex} x - z = 0 {/tex}

{tex} x - 2 z = 0 {/tex}

##### Explanation

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Q 18. The point in {tex}\mathrm {YZ}{/tex}-plane which is equidistant from three points {tex} \mathrm { A } ( 2,0,3 ) , \mathrm { B } ( 0,3,2 ) {/tex} and {tex} \mathrm { C } ( 0,0,1 ) {/tex} is

A

{tex} ( 0,3,1 ) {/tex}

{tex} ( 0,1,3 ) {/tex}

C

{tex} ( 1,3,0 ) {/tex}

D

{tex} ( 3,1,0 ) {/tex}

##### Explanation

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Q 19. The coordinates of the point {tex} R , {/tex} which divides the line segment joining {tex} P \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right) {/tex} and {tex} Q \left( x _ { 2 } , y _ { 2 } , z _ { 2 } \right) {/tex} in the ratio {tex} k: 1 , {/tex} are

A

{tex} \left( \frac { \mathrm { kx } _ { 2 } - \mathrm { x } _ { 1 } } { 1 - \mathrm { k } } , \frac { \mathrm { ky } _ { 2 } - \mathrm { y } _ { 1 } } { 1 - \mathrm { k } } , \frac { \mathrm { kz } _ { 2 } - \mathrm { z } _ { 1 } } { 1 - \mathrm { k } } \right) {/tex}

{tex} \left( \frac { \mathrm { kx } _ { 2 } + \mathrm { x } _ { 1 } } { 1 + \mathrm { k } } , \frac { \mathrm { ky } _ { 2 } + \mathrm { y } _ { 1 } } { 1 + \mathrm { k } } , \frac { \mathrm { kz } _ { 2 } + \mathrm { z } _ { 1 } } { 1 + \mathrm { k } } \right) {/tex}

C

{tex} \left( \frac { \mathrm { kx } _ { 2 } + \mathrm { x } _ { 1 } } { 1 - \mathrm { k } } , \frac { \mathrm { ky } _ { 2 } + \mathrm { y } _ { 1 } } { 1 - \mathrm { k } } , \frac { \mathrm { kz } _ { 2 } + \mathrm { z } _ { 1 } } { 1 - \mathrm { k } } \right) {/tex}

D

None of these

##### Explanation

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Q 20. The ratio, in which YZ-plane divides the line segment joining the points {tex} ( 4,8,10 ) {/tex} and {tex} ( 6,10 , - 8 ) , {/tex} is

{tex}2: 3 {/tex}(externally)

B

{tex}2: 3 {/tex} (internally)

C

{tex}1: 2 {/tex} (externally)

D

{tex}1: 2 {/tex} (internally)

##### Explanation

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Q 21. {tex}\mathbf {Assertion}{/tex} : Points {tex} ( - 4,6,10 ) , ( 2,4,6 ) {/tex} and {tex} ( 14,0 , - 2 ) {/tex} are collinear.
{tex}\mathbf {Reason} {/tex}: Point {tex} ( 14,0 , - 2 ) {/tex} divides the line segment joining by other two given points in the ratio {tex}3: 2 {/tex} internally.

A

Assertion is correct, reason is correct; reason is a correct explanation for assertion.

B

Assertion is correct, reason is correct; reason is not a correct explanation for assertion

Assertion is correct, reason is incorrect

D

Assertion is incorrect, reason is correct.

##### Explanation

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Q 22. I. The {tex} ( 0,7 , - 10 ) , ( 1,6 , - 6 ) {/tex} and {tex} ( 4,9 , - 6 ) {/tex} are the vertices of an isosceles triangle.
II. Centroid of the triangle whose vertices are {tex} \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right) , {/tex}, {tex} \left( x _ { 2 } , y _ { 2 } , z _ { 2 } \right) {/tex} and {tex} \left( x _ { 3 } , y _ { 3 } , z _ { 3 } \right) {/tex} is
{tex} \left( \frac { x _ { 1 } + x _ { 2 } + x _ { 3 } } { 3 } , \frac { y _ { 1 } + y _ { 2 } + y _ { 3 } } { 3 } , \frac { z _ { 1 } + z _ { 2 } + z _ { 3 } } { 3 } \right) {/tex}
Choose the correct option.

A

Only I is true.

B

Only II is true.

Both are true.

D

Both are false.

##### Explanation

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Q 23. Given that {tex} \mathrm { A } ( 3,2 , - 4 ) , \mathrm { B } ( 5,4 - 6 ) {/tex} and {tex} \mathrm { C } ( 9,8 , - 10 ) {/tex} are collinear. Ratio in which {tex} \mathrm { B } {/tex} divides {tex} \mathrm { AC } {/tex} is {tex} 1: \mathrm { m } . {/tex} The value of {tex} \mathrm { m } {/tex} is

2

B

3

C

4

D

5

##### Explanation

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Q 24. If the origin is the centroid of the triangle with vertices {tex} \mathrm { A } ( 2 \mathrm { a } , 2,6 ) , \mathrm { B } ( - 4,3 \mathrm { b } , - 10 ) {/tex} and {tex} \mathrm { C } ( 8,14,2 \mathrm { c } ) , {/tex} then the sum of value of {tex} \mathrm a {/tex} and {tex} \mathrm { c } {/tex} is

0

B

1

C

2

D

3

##### Explanation

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Q 25. In Δ ABC, a2(cos2B−cos2C) + b2(cos2C−cos2A) + c2 (cos2A − cos2B) is equal to

0

B

1

C

a2 + b2 + c2

D

2(a2 + b2 + c2)

##### Explanation

a2(cos2B−cos2C) + b2(cos2C−cos2A) + c2 (cos2A − cos2B)

= a2(1−sin2B−1+sin2C) + b2(1−sin2C−1+sin2A)

+ c2 (1 − sin2A − 1 + sin2B)

= a2(sin2C−sin2B) + b2(sin2A−sin2C) + c2 (sin2B−sin2A)

= k2a2(c2−b2) + k2b2(a2−c2) + k2c2 (b2 − c2)

= 0