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

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

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The locus of the points equidistant from two lines will be a parallel line.

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Important Questions on Straight Lines

EASY

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

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>Straight Lines>Basics of 2D Coordinate Geometry

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

HARD

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

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 :

EASY

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

The locus of all points that are at a distance greater than $2$ units from $(- 3, 0)$ , is

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

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:

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

Let

A = (0, 4)

and

B = (2 \cos θ, 2 \sin θ),

for some

0 < θ < \frac{π}{2} .

Let

P

divide the line segment

A B

in the ratio

2 : 3

internally. The locus of

P

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

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>Straight Lines>Basics of 2D Coordinate Geometry

A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area

C^{2}

. Then the locus of the middle point of the line is

HARD

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

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>Straight Lines>Basics of 2D Coordinate Geometry

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

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

A quadrilateral

A B C D

is divided by the diagonal

A C

into two triangles of equal areas. If

A, B, C

are respectively

(3, 4), (- 3, 6), (- 5, 1)

, then the locus of

D

HARD

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

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 :

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

If the sum of the distances from a variable point

P

to the given points

A (1, 0)

and

B (0, 1)

2

, then the locus of

P

EASY

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

The locus of a point which divides the line segment joining the point

(0, - 1)

and a point on the parabola

x^{2} = 4 y

internally in the ratio

1 : 2

, is:

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

Let

A, B

and

C

be three points in a plane. The locus of a point

P

moving such that

P A^{2} + P B^{2} = 2 P C^{2}

is a

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

The locus of a point, which moves such that the sum of squares of its distances from the points

(0, 0), (1, 0), (0, 1) (1, 1)

18

units, is a circle of diameter

d .

Then

d^{2}

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

Let

P_{1}, P_{2}

be any two points on a circle of radius

r

centred at the origin

O,

such that

∠ P_{1} O P_{2} = \frac{π}{3},

P

is the point of intersection of the tangents to the circle at

P_{1}

and

P_{2},

then the locus of the point

P,

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Basics of 2D Coordinate Geometry

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>Straight Lines>Basics of 2D Coordinate Geometry

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>Straight Lines>Basics of 2D Coordinate Geometry

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)

The locus of the points equidistant from two lines will be a parallel line.

Important Questions on Straight Lines

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