HARD

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Find the equations of the lines through the point of intersection of the lines $x - y + 1 = 0$ and $2 x - 3 y + 5 = 0$ and whose distance from the point $(3, 2)$ is $\frac{7}{5}$ .

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Important Questions on Straight Lines

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A straight line

L

at a distance of

4

units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of

60 °

with the line

x + y = 0 .

Then an equation of the line

L

is:
Note: In actual JEE Main paper, two options were correct for this question. Hence, we have changed one option.

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A line has slope

m

and

y

-intercept 4. The distance between the origin and the line is equal to

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A straight line through origin

O

meets the lines

3 y = 10 - 4 x

and

8 x + 6 y + 5 = 0

at points

A

and

B

respectively. Then,

O

divides the segment

A B

in the ratio

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Line joining the points

(0, 3)

and

(5, - 2)

is a tangent to the curve

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

, then

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

O (0, 0), A (1, 2), B (3, 4)

are the vertices of

∆ O A B .

The joint equation of the altitude and median drawn from

O

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

If we reduce

3 x + 3 y + 7 = 0

to the form

x \cos α + y \sin α = p

, then the value of

p

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Two sides of a rhombus are along the lines,

x - y + 1 = 0

and

7 x - y - 5 = 0

. If its diagonals intersect at

(- 1, - 2)

, then which one of the following is a vertex of this rhombus ?

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

If a straight line passes through the point

(- 5, 4)

and makes an intercept of length

\frac{2}{\sqrt{5}}

between the lines

x + 2 y + 1 = 0

and

x + 2 y - 1 = 0,

then the equation of that line is

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

A square, of each side

2

, lies above the

x

-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle

30 °

with the positive direction of the

x

-axis , then the sum of the

x

-coordinates of the vertices of the square is

:

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Let

b, d > 0

. The locus of all points

P (r, θ)

for which the line

O P

(where

O

is the origin) cuts the line

r \sin θ = b

Q

such that

P Q = d

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

The larger of two angles made with the

X

-axis of a straight line drawn through

(1, 2)

so that it intersects the line

x + y = 4

at a point distant

\sqrt{6} / 3

from the point

(1, 2)

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

If a straight line passing through the point

P (- 3, 4)

is such that its intercepted portion between the coordinate axes is bisected at

P

, then its equation is :

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

The tangent to the curve

y = e^{2 x}

at the point

(0, 1)

meets the

x

-axis at

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

If the perpendicular bisector of the line segment joining the points

P (1, 4)

and

Q (k, 3)

has

y

-intercept equal to

- 4

, then a value of

k

is;

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

x \cos θ + y \sin θ = p

is the normal form and

y = m x + c

is the slope - intercept form of the line

x + 2 y + 1 = 0,

then

\tan^{- 1} (\tan θ + m + c) =

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

The equation of the straight line in the normal form which is parallel to the lines

x + 2 y + 3 = 0

and

x + 2 y + 8 = 0

and dividing the distance between these two lines in the ratio

1 : 2

internally is

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

x \cos α + y \sin α = p

is the normal form of the equation of a straight line

x + \sqrt{3} y + 4 = 0

and

a, b

are respectively

X, Y -

intercepts of this line, then

\sqrt{3} π b p - 3 a α =

MEDIUM

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

The equation of perpendicular bisectors of sides

A B

and

A C

of a

∆ A B C

are

x - y + 5 = 0

and

x + 2 y = 0

respectively. If the coordinates of vertex

A

are

(1, - 2),

then equation of

B C

HARD

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Let

P S

be the median of the triangle with vertices

P (2, 2), Q (6, - 1)

and

R (7, 3)

. The equation of the line passing through

(1, - 1)

and parallel to

P S

EASY

Mathematics>Coordinate Geometry>Straight Lines>Various Forms of the Equation of a Line

Suppose that the points

(h, k), (1, 2)

and

(- 3, 4)

lie on the line

L_{1} .

If a line

L_{2}

passing through the points

(h, k)

and

(4, 3)

is perpendicular to

L_{1},

then

\frac{k}{h}

equals:

Find the equations of the lines through the point of intersection of the lines x-y+1=0 and 2x-3y+5=0 and whose distance from the point (3,2) is 75.

Important Questions on Straight Lines

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Find the equations of the lines through the point of intersection of the lines $x - y + 1 = 0$ and $2 x - 3 y + 5 = 0$ and whose distance from the point $(3, 2)$ is $\frac{7}{5}$ .