JEE Main > Coordinate Geometry

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


Q 1.    

Correct4

Incorrect-1

Find the equation to the hyperbola having its eccentricity 2 and the distance between its foci is 8 .

A

{tex} \frac { x ^ { 2 } } { 12 } - \frac { y ^ { 2 } } { 4 } = 1 {/tex}

B

{tex} \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 12 } = 1 {/tex}

{tex} \frac { x ^ { 2 } } { 8 } - \frac { y ^ { 2 } } { 2 } = 1 {/tex}

D

{tex} \frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1 {/tex}

Explanation


Q 2.    

Correct4

Incorrect-1

Two common tangents to the circle {tex} x ^ { 2 } + y ^ { 2 } = 2 a ^ { 2 } {/tex} and parabola {tex} y ^ { 2 } = 8 a x {/tex} are

A

{tex} x = \pm ( y + 2 a ) {/tex}

{tex} y = \pm ( x + 2 a ) {/tex}

C

{tex} x = \pm ( y + a ) {/tex}

D

{tex} y = \pm ( x + a ) {/tex}

Explanation



Q 3.    

Correct4

Incorrect-1

If the chord {tex} y = m x + 1 {/tex} of the circle {tex} x ^ { 2 } + y ^ { 2 } = 1 {/tex} subtends an angle of measure {tex} 45 ^ { \circ } {/tex} at the major segment of the circle then value of {tex} m {/tex} is

A

{tex} 2 \pm \sqrt { 2 } {/tex}

B

{tex} - 2 \pm \sqrt { 2 } {/tex}

{tex} - 1 \pm \sqrt { 2 } {/tex}

D

none of these

Explanation



Q 4.    

Correct4

Incorrect-1

If the pair of lines {tex} a x ^ { 2 } + 2 h x y + b y ^ { 2 } + 2 g x + 2 f y + c = 0 {/tex} intersect on the {tex} y {/tex} -axis then

{tex} 2 f g h = b g ^ { 2 } + c h ^ { 2 } {/tex}

B

{tex} b g ^ { 2 } \neq c h ^ { 2 } {/tex}

C

{tex} a b c = 2 f g h {/tex}

D

none of these

Explanation



Q 5.    

Correct4

Incorrect-1

The centres of a set of circles, each of radius {tex} 3 , {/tex} lie on the circle {tex} x ^ { 2 } + y ^ { 2 } = 25 . {/tex} The locus of any point in the set is

{tex} 4 \leq x ^ { 2 } + y ^ { 2 } \leq 64 {/tex}

B

{tex} x ^ { 2 } + y ^ { 2 } \leq 25 {/tex}

C

{tex} x ^ { 2 } + y ^ { 2 } \geq 25 {/tex}

D

{tex} 3 \leq x ^ { 2 } + y ^ { 2 } \leq 9 {/tex}

Explanation



Q 6.    

Correct4

Incorrect-1

The point of lines represented by {tex} 3 a x ^ { 2 } + 5 x y + \left( a ^ { 2 } - 2 \right) y ^ { 2 } = 0 {/tex} and {tex} \perp {/tex} to each other for

two values of {tex} a {/tex}

B

{tex} \forall {/tex} a

C

for one value of {tex} a {/tex}

D

for no values of {tex} a {/tex}

Explanation



Q 7.    

Correct4

Incorrect-1

Locus of mid point of the portion between the axes of {tex} x \cos \alpha + y \sin \alpha = p {/tex} where {tex} p {/tex} is constant is

A

{tex} x ^ { 2 } + y ^ { 2 } = \frac { 4 } { p ^ { 2 } } {/tex}

B

{tex} x ^ { 2 } + y ^ { 2 } = 4 p ^ { 2 } {/tex}

C

{tex} \frac { 1 } { x ^ { 2 } } + \frac { 1 } { y ^ { 2 } } = \frac { 2 } { p ^ { 2 } } {/tex}

{tex} \frac { 1 } { x ^ { 2 } } + \frac { 1 } { y ^ { 2 } } = \frac { 4 } { p ^ { 2 } } {/tex}

Explanation


Q 8.    

Correct4

Incorrect-1

The centre of the circle passing through {tex} ( 0,0 ) {/tex} and {tex} ( 1,0 ) {/tex} and touching the circle {tex} x ^ { 2 } + y ^ { 2 } = 9 {/tex} is

A

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

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

C

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

D

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

Explanation



Q 9.    

Correct4

Incorrect-1

The equation of a circle with origin as a centre and passing through equilateral triangle whose median is of length {tex} 3 a {/tex} is

A

{tex} x ^ { 2 } + y ^ { 2 } = 9 a ^ { 2 } {/tex}

B

{tex} x ^ { 2 } + y ^ { 2 } = 16 a ^ { 2 } {/tex}

{tex} x ^ { 2 } + y ^ { 2 } = 4 a ^ { 2 } {/tex}

D

{tex} x ^ { 2 } + y ^ { 2 } = a ^ { 2 } {/tex}

Explanation


Q 10.    

Correct4

Incorrect-1

A triangle with vertices {tex} ( 4,0 ) , ( - 1 , - 1 ) , ( 3,5 ) {/tex} is

isosceles and right angled

B

isosceles but not right angled

C

right angled but not isosceles

D

neither right angled nor isosceles

Explanation



Q 11.    

Correct4

Incorrect-1

The foci of the ellipse {tex} \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 {/tex} and the hyperbola {tex} \frac { x ^ { 2 } } { 144 } - \frac { y ^ { 2 } } { 81 } = \frac { 1 } { 25 } {/tex} coincide. Then the value of {tex} b ^ { 2 } {/tex} is

A

5

7

C

9

D

1

Explanation




Q 12.    

Correct4

Incorrect-1

The normal at the point {tex} \left( b t _ { 1 } ^ { 2 } , 2 b t _ { 1 } \right) {/tex} on a parabola meets the parabola again in the point {tex} \left( b t _ { 2 } ^ { 2 } , 2 b t _ { 2 } \right) , {/tex} then

A

{tex} t _ { 2 } = - t _ { 1 } + \frac { 2 } { t _ { 1 } } {/tex}

B

{tex} t _ { 2 } = t _ { 1 } - \frac { 2 } { t _ { 1 } } {/tex}

C

{tex} t _ { 2 } = t _ { 1 } + \frac { 2 } { t _ { 1 } } {/tex}

{tex} t _ { 2 } = - t _ { 1 } - \frac { 2 } { t _ { 1 } } {/tex}

Explanation


Q 13.    

Correct4

Incorrect-1

If the two circles {tex} ( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 } {/tex} and {tex} x ^ { 2 } + y ^ { 2 } - 8 x + 2 y + 8 = 0 {/tex} intersect in two distinct points, then

A

{tex} r< 2 {/tex}

B

{tex} r = 2 {/tex}

C

{tex} r > 2 {/tex}

{tex} 2 < r < 8 {/tex}

Explanation


Q 14.    

Correct4

Incorrect-1

The lines {tex} 2 x - 3 y = 5 {/tex} and {tex} 3 x - 4 y = 7 {/tex} are diameters of a circle having area as 154 sq. units. Then the equation of the circle is

A

{tex} x ^ { 2 } + y ^ { 2 } + 2 x - 2 y = 47 {/tex}

{tex} x ^ { 2 } + y ^ { 2 } - 2 x + 2 y = 47 {/tex}

C

{tex} x ^ { 2 } + y ^ { 2 } - 2 x + 2 y = 62 {/tex}

D

{tex} x ^ { 2 } + y ^ { 2 } + 2 x - 2 y = 62 {/tex}

Explanation


Q 15.    

Correct4

Incorrect-1

A square of side {tex} a {/tex} lies above the {tex} x {/tex} -axis and has one vertex at the origin. The side passing through the origin makes an angle {tex} \alpha ( 0 < \alpha < \pi / 4 ) {/tex} with the positive direction of {tex} x {/tex} -axis. The equation of its diagonal not passing through the origin is

A

{tex} y ( \cos \alpha + \sin \alpha ) + x ( \sin \alpha - \cos \alpha ) = a {/tex}

B

{tex} y ( \cos \alpha + \sin \alpha ) + x ( \sin \alpha + \cos \alpha ) = a {/tex}

{tex} y ( \cos \alpha + \sin \alpha ) + x ( \cos \alpha - \sin \alpha ) = a {/tex}

D

{tex} y ( \cos \alpha - \sin \alpha ) - x ( \sin \alpha - \cos \alpha ) = a {/tex}

Explanation



Q 16.    

Correct4

Incorrect-1

If the pairs of straight lines {tex} x ^ { 2 } - 2 p x y - y ^ { 2 } = 0 {/tex} and {tex} x ^ { 2 } - 2 q x y - y ^ { 2 } = 0 {/tex} be such that each pair bisects the angle between the other pair, then

A

{tex} p = - q {/tex}

B

{tex} p q = 1 {/tex}

{tex} p q = - 1 {/tex}

D

{tex} p = q {/tex}

Explanation


Q 17.    

Correct4

Incorrect-1

Locus of centroid of the triangle whose vertices are {tex} ( a \text { cost, } a \text { sint } ) , ( b \text { sint, } - b \text { cost } ) {/tex} and {tex} ( 1,0 ) , {/tex} where {tex} t {/tex} is a parameter, is

{tex} ( 3 x - 1 ) ^ { 2 } + ( 3 y ) ^ { 2 } = a ^ { 2 } + b ^ { 2 } {/tex}

B

{tex} ( 3 x + 1 ) ^ { 2 } + ( 3 y ) ^ { 2 } = a ^ { 2 } + b ^ { 2 } {/tex}

C

{tex} ( 3 x + 1 ) ^ { 2 } + ( 3 y ) ^ { 2 } = a ^ { 2 } - b ^ { 2 } {/tex}

D

{tex} ( 3 x - 1 ) ^ { 2 } + ( 3 y ) ^ { 2 } = a ^ { 2 } - b ^ { 2 } {/tex}

Explanation



Q 18.    

Correct4

Incorrect-1

If the equation of the locus of point equidistant from the points {tex} \left( a _ { 1 } , b _ { 1 } \right) {/tex} and {tex} \left( a _ { 2 } , b _ { 2 } \right) {/tex} is {tex} \left( a _ { 1 } - a _ { 2 } \right) x + \left( b _ { 1 } - b _ { 2 } \right) y + c = 0 , {/tex} then {tex} c = {/tex}

A

{tex} a _ { 1 } ^ { 2 } - a _ { 2 } ^ { 2 } + b _ { 1 } ^ { 2 } - b _ { 2 } ^ { 2 } {/tex}

B

{tex} \frac { 1 } { 2 } \left( a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + b _ { 1 } ^ { 2 } + b _ { 2 } ^ { 2 } \right) {/tex}

C

{tex} \sqrt { \left( a _ { 1 } ^ { 2 } + b _ { 1 } ^ { 2 } - a _ { 2 } ^ { 2 } - b _ { 2 } ^ { 2 } \right) } {/tex}

{tex} \frac { 1 } { 2 } \left( a _ { 2 } ^ { 2 } + b _ { 2 } ^ { 2 } - a _ { 1 } ^ { 2 } - b _ { 1 } ^ { 2 } \right) {/tex}

Explanation



Q 19.    

Correct4

Incorrect-1

A point on the parabola {tex} y ^ { 2 } = 18 x {/tex} at which the ordinate increases at twice the rate of the abscissa is

A

{tex} \left( \frac { - 9 } { 8 } , \frac { 9 } { 2 } \right) {/tex}

B

{tex} ( 2 , - 4 ) {/tex}

C

{tex} ( 2,4 ) {/tex}

{tex} \left( \frac { 9 } { 8 } , \frac { 9 } { 2 } \right) {/tex}

Explanation


Q 20.    

Correct4

Incorrect-1

The normal to the curve {tex} x = a ( 1 + \cos \theta ) , y = a \sin \theta {/tex} at {tex} \theta {/tex} always passes through the fixed point

A

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

B

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

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

D

{tex} ( a , a ) {/tex}

Explanation


Q 21.    

Correct4

Incorrect-1

Let {tex} A ( 2 , - 3 ) {/tex} and {tex} B ( - 2,1 ) {/tex} be vertices of a triangle {tex} A B C {/tex}. If the centroid of this triangle moves on the line {tex} 2 x + 3 y = 1 , {/tex} then the locus of the vertex {tex} C {/tex} is the line

A

{tex} 3 x + 2 y = 5 {/tex}

B

{tex} 2 x - 3 y = 7 {/tex}

{tex} 2 x + 3 y = 9 {/tex}

D

{tex} 3 x - 2 y = 3 {/tex}

Explanation


Q 22.    

Correct4

Incorrect-1

The equation of the straight line passing through the point {tex} ( 4,3 ) {/tex} and making intercepts on the coordinate axes whose sum is {tex} - 1 {/tex} is

A

{tex} \frac { x } { 2 } + \frac { y } { 3 } = 1 {/tex} and {tex} \frac { x } { 2 } + \frac { y } { 1 } = 1 {/tex}

B

{tex} \frac { x } { 2 } - \frac { y } { 3 } = - 1 {/tex} and {tex} \frac { x } { - 2 } + \frac { y } { 1 } = 1 {/tex}

C

{tex} \frac { x } { 2 } + \frac { y } { 3 } = - 1 {/tex} and {tex} \frac { x } { - 2 } + \frac { y } { 1 } = - 1 {/tex}

{tex} \frac { x } { 2 } - \frac { y } { 3 } = 1 {/tex} and {tex} \frac { x } { - 2 } + \frac { y } { 1 } = 1 {/tex}

Explanation



Q 23.    

Correct4

Incorrect-1

If the sum of the slopes of the lines given by {tex} x ^ { 2 } - 2 c x y - 7 y ^ { 2 } = 0 {/tex} is four times their product, then {tex} c {/tex} has the value

{tex}2{/tex}

B

{tex} - 1 {/tex}

C

{tex} 1 {/tex}

D

{tex} - 2 {/tex}

Explanation


Q 24.    

Correct4

Incorrect-1

If one of the lines given by {tex} 6 x ^ { 2 } - x y + 4 c y ^ { 2 } = 0 {/tex} is {tex} 3 x + 4 y = 0 , {/tex} then {tex} c {/tex} equals

A

3

B

-1

C

1

-3

Explanation



Q 25.    

Correct4

Incorrect-1

If a circle passes through the point {tex} ( a , b ) {/tex} and cuts the circle {tex} x ^ { 2 } + y ^ { 2 } = 4 {/tex} orthogonally, then the locus of its centre is

A

{tex} 2 a x - 2 b y + \left( a ^ { 2 } + b ^ { 2 } + 4 \right) = 0 {/tex}

{tex} 2 a x + 2 b y - \left( a ^ { 2 } + b ^ { 2 } + 4 \right) = 0 {/tex}

C

{tex} 2 a x + 2 b y + \left( a ^ { 2 } + b ^ { 2 } + 4 \right) = 0 {/tex}

D

{tex} 2 a x - 2 b y - \left( a ^ { 2 } + b ^ { 2 } + 4 \right) = 0 {/tex}

Explanation