The locus of the point of intersection of the lines 2 x - y + 4 2 k = 0 and 2 k x + k y - 4 2 = 0 ( k is any non-zero real parameter) is

a hyperbola with length of its transverse axis 8 2

an ellipse whose eccentricity is 1 3

a hyperbola whose eccentricity is 3

an ellipse with length of its major axis 8 2

Let a and b be any two numbers satisfying 1 a 2 + 1 b 2 = 1 4 . Then, the foot of perpendicular from the origin on the variable line x a + y b = 1 lies on :

A hyperbola with each semi-axis = 2 .

A hyperbola with each semi-axis = 2

HARD

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Let $A$ lies on $3 x - 4 y + 1 = 0,$ $B$ lies on $4 x + 3 y - 7 = 0$ and $C$ is $(- 2, 5) .$ If $A B C D$ is a rhombus, then the locus of $D$ is a conic whose length of the latus rectum is equal to

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Important Questions on Point and Straight Line

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

A point

P

moves on the line

2 x - 3 y + 4 = 0 .

Q (1, 4)

and

R (3, - 2)

are fixed points, then the locus of the centroid of

Δ P Q R

is a line:

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

If a circle of radius

R

passes through the origin

O

and intersects the coordinate axes at

A

and

B

, then the locus of the foot of perpendicular from

O

A B

is :

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Let

A B C D

be a square of side length

1 .

Let

P, Q, R, S

be points in the interiors of the sides

A D, B C, A B, C D

respectively, such that

P Q

and

R S

intersect at right angles. If

P Q = \frac{3 \sqrt{3}}{4},

then

R S

equals

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Consider a triangle

A B C

in the

x y

- plane with vertices

A = (0, 0), B = (1,1) a n d C = (9, 1)

. If the line

x = a

divides the triangle into two parts of equal area, then

a

equals

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Let

A = (a_{1}, a_{2})

and

B = (b_{1}, b_{2})

be two points in the plane with integer coordinates. Which one of the following is not a possible value of the distance between

A

and

B

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Two line segments

A B

and

C D

are constrained to move along the

x

and

y

axes, respectively, in such a way that the points

A, B, C, D

are concyclic. If

A B = a a n d C D = b

, then the locus of the center of the circle passing through

A, B, C, D

in polar coordinates is

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

The coordinate of the point dividing internally the line joining the points

(4, - 2)

and

(8, 6)

in the ratio

7 : 5

EASY

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

A straight line through a fixed point

(2, 3)

intersects the coordinate axes at distinct points

P

and

Q

. If

O

is the origin and the rectangle

O P R Q

is completed, then the locus of

R

is:

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

The locus of the point of intersection of the lines

\sqrt{2} x - y + 4 \sqrt{2} k = 0

and

\sqrt{2} k x + k y - 4 \sqrt{2} = 0

(

k

is any non-zero real parameter) is

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

A wall is inclined to the floor at an angle of $135 ° .$ A ladder of length $l$ is resting on the wall. As the ladder slides down, its mid-point traces an arc of an ellipse. Then the area of the ellipse is
Question Image

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

If the sum of distances from a point

P

on two mutually perpendicular straight lines is

1

unit, then the locus of

P

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Let

S

be the set of all triangles in the

x y

-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in

S

has area

50

sq. units, then the number of elements in the set

S

is:

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Let

C

be the circle with centre

(0, 0)

and radius

3 unit

. The equation of the locus of the mid points of the chords of the circle

C

that subtend an angle of

\frac{2 π}{3}

at its centre, is:

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Let

A B C D

be a square of side length

1

, and

Γ

a circle passing through

B

and

C

, and touching

A D

. The radius of

Γ

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Let

a

and

b

be any two numbers satisfying

\frac{1}{a^{2}} + \frac{1}{b^{2}} = \frac{1}{4} .

Then, the foot of perpendicular from the origin on the variable line

\frac{x}{a} + \frac{y}{b} = 1

lies on :

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Let

A (1, 0), B (6, 2)

and

C (\frac{3}{2}, 6)

be the vertices of a triangle

A B C .

P

is a point inside the triangle

A B C

such that the triangles

A P C, A P B

and

B P C

have equal areas, then the length of the line segment

P Q,

where

Q

is the point

(- \frac{7}{6}, - \frac{1}{3}),

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Find the ratio in which line $3 x + 2 y = 17$ divides the line segment joined by points $(2, 5)$ and $(5, 2) .$

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

In a

∆ A B C

, medians,

A D

and

B E

are drawn. If

A D = 4

∠ D A B = \frac{π}{6} a n d ∠ A B E = \frac{π}{3}

, then the area of the

∆ A B C

HARD

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Let

B C

be a fixed line segment in the plane. The locus of a point

A

such that the triangle

A B C

is isosceles, is (with finitely many possible exceptional points)

MEDIUM

Mathematics>Coordinate Geometry>Point and Straight Line>Point in Cartesian Plane

Locus of the image of the point $(2, 3)$ in the line $(2 x - 3 y + 4) + k (x - 2 y + 3) = 0, k \in R,$ is a

Let A lies on 3x-4y+1=0, B lies on 4x+3y-7=0 and C is -2,5. If ABCD is a rhombus, then the locus of D is a conic whose length of the latus rectum is equal to

Important Questions on Point and Straight Line

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Let $A$ lies on $3 x - 4 y + 1 = 0,$ $B$ lies on $4 x + 3 y - 7 = 0$ and $C$ is $(- 2, 5) .$ If $A B C D$ is a rhombus, then the locus of $D$ is a conic whose length of the latus rectum is equal to