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JEE Advanced > Coordinate Geometry

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

Q 1.

Correct4

Incorrect-1

The equation {tex} \frac { x ^ { 2 } } { 1 - r } - \frac { y ^ { 2 } } { 1 + r } = 1 , \quad r > 1 {/tex} represents

A

an ellipse

B

a hyperbola

C

a circle

none of these

Explanation

Q 2.

Correct4

Incorrect-1

Each of the four inequalties given below defines a region in the {tex} x y {/tex} plane. One of these four regions does not have the following property. For any two points {tex} \left( x _ { 1 } , y _ { 1 } \right) {/tex} and {tex} \left( x _ { 2 } , y _ { 2 } \right) {/tex} in the region, the point {tex} \left( \frac { x _ { 1 } + x _ { 2 } } { 2 } , \frac { y _ { 1 } + y _ { 2 } } { 2 } \right) {/tex} is also in the region. The inequality defining this region is

A

{tex} x ^ { 2 } + 2 y ^ { 2 } \leq 1 {/tex}

B

{tex} \operatorname { Max } \{\ | x | , | y | \ \} \leq 1 {/tex}

{tex} x ^ { 2 } - y ^ { 2 } \leq 1 {/tex}

D

{tex} y ^ { 2 } - x \leq 0 {/tex}

Explanation



Q 3.

Correct4

Incorrect-1

Let {tex} E {/tex} be the ellipse {tex} \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1 {/tex} and {tex} C {/tex} be the circle {tex} x ^ { 2 } + y ^ { 2 } = 9 . {/tex} Let {tex} P {/tex} and {tex} Q {/tex} be the points {tex} ( 1,2 ) {/tex} and {tex} ( 2,1 ) {/tex}, respectively, Then

A

{tex} Q {/tex} lies inside {tex} C {/tex} but outside {tex} E {/tex}

B

{tex} Q {/tex} lies outside both {tex} C {/tex} and {tex} E {/tex}

C

{tex} P {/tex} lies inside both {tex} C {/tex} and {tex} E {/tex}

{tex} P {/tex} lies inside {tex} C {/tex} but outside {tex} E {/tex}

Explanation

Q 4.

Correct4

Incorrect-1

Consider a circle with its centre lying on the focus of the parabola {tex} y ^ { 2 } = 2 p x {/tex} such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is

{tex} \left( \frac { p } { 2 } , p \right) {/tex} or {tex} \left( \frac { p } { 2 } , - p \right) {/tex}

B

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

C

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

D

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

Explanation


Q 5.

Correct4

Incorrect-1

The radius of the circle passing through the foci of the ellipse {tex} \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1 , {/tex} and having its centre at {tex} ( 0,3 ) {/tex} is

4

B

3

C

{tex} \sqrt { \frac { 1 } { 2 } } {/tex}

D

{tex} \frac { 7 } { 2 } {/tex}

Explanation

Q 6.

Correct4

Incorrect-1

Let {tex} P ( \text { a sec } \theta , b \text { tan } \theta ) {/tex} and {tex} Q ( a \sec \phi , b \tan \phi ) , {/tex} where {tex} \theta + \phi = \pi / 2 , {/tex} be two points on the hyperbola {tex} \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1. {/tex} If {tex} ( h , k ) {/tex} is the point of intersection of the normals at {tex} P {/tex} and {tex} Q , {/tex} then {tex} k {/tex} is equal to

A

{tex} \frac { a ^ { 2 } - b ^ { 2 } } { a } {/tex}

B

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

C

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

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

Explanation


Q 7.

Correct4

Incorrect-1

If {tex} x = 9 {/tex} is the chord of contact of the hyperbola {tex} x ^ { 2 } - y ^ { 2 } = 9 {/tex} then the equation of the corresponding pair of tangents is

A

{tex} 9 x ^ { 2 } - 8 y ^ { 2 } + 18 x - 9 = 0 {/tex}

{tex} 9 x ^ { 2 } - 8 y ^ { 2 } - 18 x + 9 = 0 {/tex}

C

{tex} 9 x ^ { 2 } - 8 y ^ { 2 } - 18 x - 9 = 0 {/tex}

D

{tex} 9 x ^ { 2 } - 8 y ^ { 2 } + 18 x + 9 = 0 {/tex}

Explanation

Q 8.

Correct4

Incorrect-1

If {tex} x + y = \mathrm { k } {/tex} is normal to {tex} \mathrm { y } ^ { 2 } = 12 x , {/tex} then {tex} k {/tex} is

A

{tex} 3 {/tex}

{tex} 9 {/tex}

C

{tex} - 9 {/tex}

D

{tex} - 3 {/tex}

Explanation

Q 9.

Correct4

Incorrect-1

If the line {tex} x - 1 = 0 {/tex} is the directrix of the parabola {tex} y ^ { 2 } - k x + 8 = 0 , {/tex} then one of the values of {tex} k {/tex} is

A

{tex} 1 / 8 {/tex}

B

{tex} 8 {/tex}

{tex} 4 {/tex}

D

{tex} 1 / 4 {/tex}

Explanation


Q 10.

Correct4

Incorrect-1

The equation of the directrix of the parabola
{tex} \quad y ^ { 2 } + 4 y + 4 x + 2 = 0 {/tex} is

A

{tex} x = - 1 {/tex}

B

{tex} x = 1 {/tex}

C

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

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

Explanation

Q 11.

Correct4

Incorrect-1

If {tex} a > 2 b > 0 {/tex} then the positive value of {tex} m {/tex} for which {tex} y = m x - b \sqrt { 1 + m ^ { 2 } } {/tex} is a common tangent to {tex} x ^ { 2 } + y ^ { 2 } = b ^ { 2 } {/tex} and {tex} ( x - a ) ^ { 2 } + y ^ { 2 } = b ^ { 2 } {/tex} is

{tex} \frac { 2 b } { \sqrt { a ^ { 2 } - 4 b ^ { 2 } } } {/tex}

B

{tex} \frac { \sqrt { a ^ { 2 } - 4 b ^ { 2 } } } { 2 b } {/tex}

C

{tex} \frac { 2 b } { a - 2 b } {/tex}

D

{tex} \frac { b } { a - 2 b } {/tex}

Explanation


Q 12.

Correct4

Incorrect-1

The angle between the tangents drawn from the point {tex} ( 1,4 ) {/tex} to the parabola {tex} y ^ { 2 } = 4 x {/tex} is

A

{tex} \pi / 6 {/tex}

B

{tex} \pi / 4 {/tex}

{tex} \pi / 3 {/tex}

D

{tex} \pi / 2 {/tex}

Explanation

Q 13.

Correct4

Incorrect-1

The minimum area of triangle formed by the tangent to the {tex} \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1\ \& {/tex} coordinate axes is

{tex} a b {/tex} sq. units

B

{tex} \frac { a ^ { 2 } + b ^ { 2 } } { 2 } {/tex} sq. units

C

{tex} \frac { ( a + b ) ^ { 2 } } { 2 } {/tex} sq. units

D

{tex} \frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 } {/tex} sq. units

Explanation


Q 14.

Correct4

Incorrect-1

Tangent to the curve {tex} y = x ^ { 2 } + 6 {/tex} at a point {tex} ( 1,7 ) {/tex} touches the circle {tex} x ^ { 2 } + y ^ { 2 } + 16 x + 12 y + c = 0 {/tex} at a point {tex} Q . {/tex} Then the coordinates of {tex} Q {/tex} are

A

{tex} ( - 6 , - 11 ) {/tex}

B

{tex} ( - 9 , - 13 ) {/tex}

C

{tex} ( - 10 , - 15 ) {/tex}

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

Explanation


Q 15.

Correct4

Incorrect-1

A hyperbola, having the transverse axis of length {tex} 2 \sin \theta , {/tex} is confocal with the ellipse {tex} 3 x ^ { 2 } + 4 y ^ { 2 } = 12 . {/tex} Then its equation is

{tex} x ^ { 2 } \mathrm {cosec} ^ { 2 } \theta - y ^ { 2 } \sec ^ { 2 } \theta = 1 {/tex}

B

{tex} x ^ { 2 } \sec ^ { 2 } \theta - y ^ { 2 } \mathrm {cosec} ^ { 2 } \theta = 1 {/tex}

C

{tex} x ^ { 2 } \sin ^ { 2 } \theta - y ^ { 2 } \cos ^ { 2 } \theta = 1 {/tex}

D

{tex} x ^ { 2 } \cos ^ { 2 } \theta - y ^ { 2 } \sin ^ { 2 } \theta = 1 {/tex}

Explanation


Q 16.

Correct4

Incorrect-1

The line passing through the extremity {tex} A {/tex} of the major axis and extremity {tex} B {/tex} of the minor axis of the ellipse \[ x ^ { 2 } + 9 y ^ { 2 } = 9 \]
meets its auxiliary circle at the point {tex} M . {/tex} Then the area of the triangle with vertices at {tex} A , M {/tex} and the origin {tex} O {/tex} is

A

{tex} \frac { 31 } { 10 } {/tex}

B

{tex} \frac { 29 } { 10 } {/tex}

C

{tex} \frac { 21 } { 10 } {/tex}

{tex} \frac { 27 } { 10 } {/tex}

Explanation



Q 17.

Correct4

Incorrect-1

The points {tex} ( - a , - b ) , ( 0,0 ) , ( a , b ) {/tex} and {tex} \left( a ^ { 2 } , a b \right) {/tex} are:

Collinear

B

Vertices of a parallelogram

C

Vertices of a rectangle

D

None of these

Explanation

Q 18.

Correct4

Incorrect-1

The point {tex} ( 4,1 ) {/tex} undergoes the following three transformations successively.
(i) Reflection about the line {tex} y = x {/tex}
(ii) Translation through a distance 2 units along the positive direction of {tex} x {/tex} -axis.
(iii) Rotation through an angle {tex} p / 4 {/tex} about the origin in the counter clockwise direction.
Then the final position of the point is given by the coordinates.

A

{tex} \left( \frac { 1 } { \sqrt { 2 } } , \frac { 7 } { \sqrt { 2 } } \right) {/tex}

B

{tex} ( - \sqrt { 2 } , 7 \sqrt { 2 } ) {/tex}

{tex} \left( - \frac { 1 } { \sqrt { 2 } } , \frac { 7 } { \sqrt { 2 } } \right) {/tex}

D

{tex} ( \sqrt { 2 } , 7 \sqrt { 2 } ) {/tex}

Explanation


Q 19.

Correct4

Incorrect-1

The straight lines {tex} x + y = 0,3 x + y - 4 = 0 , x + 3 y - 4 = 0 {/tex} form a triangle which is

isosceles

B

equilateral

C

right angled

D

none of these

Explanation

Q 20.

Correct4

Incorrect-1

If {tex} P = ( 1,0 ) , Q = ( - 1,0 ) {/tex} and {tex} R = ( 2,0 ) {/tex} are three given points, then locus of the point {tex} S {/tex} satisfying the relation {tex} S Q ^ { 2 } + S R ^ { 2 } = 2 S P ^ { 2 } , {/tex} is

a straight line parallel to x-axis

B

a circle passing through the origin

C

a circle with the centre at the origin

D

a straigth line parallel to y-axis

Explanation

Q 21.

Correct4

Incorrect-1

Line L has intercepts a and b on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line {tex} L {/tex} has intercepts {tex} p {/tex} and {tex} q , {/tex} then

A

{tex} a ^ { 2 } + b ^ { 2 } = p ^ { 2 } + q ^ { 2 } {/tex}

{tex} \frac { 1 } { a ^ { 2 } } + \frac { 1 } { b ^ { 2 } } = \frac { 1 } { p ^ { 2 } } + \frac { 1 } { q ^ { 2 } } {/tex}

C

{tex} a ^ { 2 } + p ^ { 2 } = b ^ { 2 } + q ^ { 2 } {/tex}

D

{tex} \frac { 1 } { a ^ { 2 } } + \frac { 1 } { p ^ { 2 } } = \frac { 1 } { b ^ { 2 } } + \frac { 1 } { q ^ { 2 } } {/tex}

Explanation


Q 22.

Correct4

Incorrect-1

If the sum of the distances of a point from two perpendicular lines in a plane is {tex} 1 , {/tex} then its locus is

square

B

circle

C

straight line

D

two intersecting lines

Explanation


Q 23.

Correct4

Incorrect-1

The locus of a variable point whose distance from {tex} ( - 2,0 ) {/tex} is {tex} 2 / 3 {/tex} times its distance from the line {tex} x = - \frac { 9 } { 2 } {/tex} is

ellipse

B

parabola

C

hyperbola

D

none of these

Explanation

Q 24.

Correct4

Incorrect-1

The equations to a pair of opposite sides of parallelogram are {tex} x ^ { 2 } - 5 x + 6 = 0 {/tex} and {tex} y ^ { 2 } - 6 y + 5 = 0 {/tex}, the equations to its diagonals are

A

{tex} x + 4 y = 13 , y = 4 x - 7 {/tex}

B

{tex} 4 x + y = 13,4 y = x - 7 {/tex}

{tex} 4 x + y = 13 , y = 4 x - 7 {/tex}

D

{tex} y - 4 x = 13 , y + 4 x = 7 {/tex}

Explanation

Q 25.

Correct4

Incorrect-1

The orthocentre of the triangle formed by the lines {tex} x y = 0 {/tex} and {tex} x + y = 1 {/tex} is

A

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

B

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

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

D

{tex} \left( \frac { 1 } { 4 } , \frac { 1 } { 4 } \right) {/tex}

Explanation