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The Locus of centers of the circles, possessing the same area and having $3 x - 4 y + 4 = 0$ and $6 x - 8 y - 7 = 0$ as their common tangent, is

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Important Questions on Straight Line

EASY

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

Distance between the lines

4 x + 3 y = 6

and

4 x + 3 y = 11

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

If the points

(1, 1, λ) & (- 3, 0, 1)

, are equidistant from the plane,

3 x + 4 y - 12 z + 13 = 0,

then

λ

satisfies the equation:

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

Let

S

be a subset of the plane defined by

S = \{(x, y) : | x | + 2 | y | = 1\}

. Then, the radius of the smallest circle with centre at the origin and having non-empty intersection with

S

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

2 x + 3 y + 4 = 0

is the perpendicular bisector of the line segment joining the points

A (1, 2)

and

B (α, β)

, then the value of

13 α + 13 β

equals_____

EASY

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

The distance between two parallel lines

5 x - 12 y + 2 = 0

and

5 x - 12 y - 3 = 0

is given by

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

p

and

q

are the lengths of the perpendiculars from the origin on the lines,

x cosec α - y \sec α = k \cot 2 α

and

x \sin α + y \cos α = k \sin 2 α

respectively, then

k^{2}

is equal to :

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

Let

a \neq 0, b \neq 0, c

be three real numbers and

L (p, q) = \frac{a p + b q + c}{\sqrt{a^{2} + b^{2}}}, \forall p, q \in R

. If

L (\frac{2}{3}, \frac{1}{3}) + L (\frac{1}{3}, \frac{2}{3}) + L (2, 2) = 0

, then the line

a x + b y + c = 0

always passes through the fixed point

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

Let

B

and

C

be the two points on the line

y + x =0

such that

B

and

C

are symmetric with respect to the origin. Suppose

A

is a point on

y - 2 x = 2

such that

∆ A B C

is an equilateral triangle. Then, the area of the

∆ A B C

EASY

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

The length of perpendicular from the point

(a \cos α, a \sin α)

upon the straight line

y = x \tan α + c, c > 0

HARD

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

A ray of light is projected from the origin at angle of

60 °

with the positive direction of

x

-axis towards the line,

y = 2,

which gets reflected from the point,

(α, 2) .

Then the distance of the reflected ray of light from the point

(2, 2)

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

The equation of the base of an equilateral triangle is

x + y = 2

and one vertex is

(2, - 1)

, then the length of the side of the triangle is

EASY

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

Distance between two parallel lines

y = 2 x + 4

and

y = 2 x - 1

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

Let

R

be the point

(3, 7)

and let

P

and

Q

be two points on the line

x + y = 5

such that

P Q R

is an equilateral triangle. Then the area of

∆ P Q R

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

m

and

n

are the lengths of the perpendicular from the origin to the straight lines whose equations are

x \cot θ - y = 2 \cos θ

and

4 x + 3 y = - \sqrt{5} \cos 2 θ

(θ \in (0, π)),

respectively, then the value of

m^{2} + 5 n^{2}

EASY

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

If the line

x - y + 1 = 0

cuts the lines

2 x + 2 y + 3 = 0

and

3 x + 3 y + 2 = 0

at the points

A

and

B

respectively, then

A B =

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

Two sides of a parallelogram are along the lines,

x + y = 3

and

x - y + 3 = 0 .

If its diagonals intersect at

(2, 4),

then one of its vertex is:

HARD

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

If the straight line

2 x - 3 y + 17 = 0

is perpendicular to the line passing through the points

(7, 17)

and

(15, β)

, then

β

equals :

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

The length of the perpendicular from

(1, - 2)

to the line

12 x + 5 y + 63 = 0

MEDIUM

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

If $Q (0, - 1, - 3)$ is the image of the point $P$ in the plane $3 x - y + 4 z = 2$ and $R$ is the point $(3, - 1, - 2)$ , then the area (in sq. units) of $Δ P Q R$ is

Question Image

HARD

Mathematics>Coordinate Geometry>Straight Line>Distance of a Point From a Line

If a line

L

is perpendicular to the line

5 x - y = 1,

and the area of the triangle formed by the line

L

and the coordinate axes is

5

sq units, then the distance of the line

L

from the line

x + 5 y = 0

The Locus of centers of the circles, possessing the same area and having 3x-4y+4=0 and 6x-8y-7=0 as their common tangent, is

Important Questions on Straight Line

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The Locus of centers of the circles, possessing the same area and having $3 x - 4 y + 4 = 0$ and $6 x - 8 y - 7 = 0$ as their common tangent, is