If a variable line drawn through the intersection of the lines x 3 + y 4 = 1 and x 4 + y 3 = 1 , meets the coordinate axes at A and B , A ≠ B , then the locus of the midpoint of AB is:

14 x + y 2 - 97 x + y + 168 = 0

EASY

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If the lines $2 x - 3 y + 5 = 0, 9 x - 5 y + 14 = 0$ and $3 x - 7 y + λ = 0$ are concurrent, then the value of $λ$ is equal to

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Important Questions on The Straight Line

MEDIUM

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

Two vertices of a triangle are

(0, 2)

and

(4, 3) .

If its orthocenter is at the origin, then its third vertex lies in which quadrant?

EASY

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

If the lines

L_{1} \equiv 2 x + y + 3 = 0, L_{2} \equiv k x + 2 y - 3 = 0

and

L_{3} \equiv 3 x - 2 y + 1 = 0

are concurrent then the cosine of the acute angle between the lines

L_{2} = 0

and

2 x - 5 y + 7 = 0

MEDIUM

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

If the family of straight lines

a x + b y + c = 0

, where

2 a + 3 b = 4 c

is concurrent at the point

P (l, m)

, then the foot of the perpendicular drawn from

P

to the line

x + y + 1 = 0

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

A straight line

L

through the point

(3, - 2)

is inclined at an angle of

60 °

to the line

\sqrt{3} x + y = 1 .

L

also intersects the

X

-axis, then the equation of

L

is:

EASY

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

Let each of the equations

x^{2} + 2 x y + a y^{2} = 0

and

a x^{2} + 2 x y + y^{2} = 0

represent two straight lines passing through the origin. If they have a common line, then the other two lines are given by

MEDIUM

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

The number of values of $k$ , for which the system of equations :

$(k + 1) x + 8 y = 4 k$

$k x + (k + 3) y = 3 k - 1$

has no solution, is :

EASY

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

Consider the set of all lines

p x + q y + r = 0

such that

3 p + 2 q + 4 r = 0 .

Which one of the following statements is true?

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

The line which is parallel to

x

-axis and crosses the curve

y = \sqrt{x}

at an angle

45 °

, is

MEDIUM

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

Let

C

be the centroid of the triangle with vertices

(3, - 1), (1, 3)

and

(2, 4) .

Let

P

be the point of intersection of the lines

x + 3 y - 1 = 0

and

3 x - y + 1 = 0 .

Then the line passing through the points

C

and

P

also passes through the point

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

The line

(K + 1)^{2} x + K y - 2 K^{2} - 2 = 0

passes through a point regardless of the value of

K

. Which of the following is the equation of the line with slope

2

and passing through that point?

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

Let

L

be the line passing through the point

P (1, 2)

such that its intercepted segment between the co-ordinate axes is bisected at

P

. If

L_{1}

is the line perpendicular to

L

and passing through the point

(- 2, 1)

, then the point of intersection of

L

and

L_{1}

MEDIUM

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

The equation of the line passing through the point of intersection of lines

2 x - y + 2 = 0

and

x + y + 4 = 0

and the point

(5, - 2)

MEDIUM

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

Let

a, b, c

and

d

be non-zero numbers. If the point of intersection of the lines

4 a x + 2 a y + c = 0 & 5 b x + 2 b y + d = 0

lies in the fourth quadrant and is equidistant from the two axes then

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

The tangent to the curve

y = x^{2} - 5 x + 5

, parallel to the line

2 y = 4 x + 1

, also passes through the point :

EASY

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

The equation of the line passing through

(a, b)

and parallel to the line

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

MEDIUM

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

Slope of a line passing through

P (2, 3)

and intersecting the line

x + y = 7

at a distance of

4

units from

P,

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

The point

(2, 1)

is translated parallel to the line

L : x - y = 4

2 \sqrt{3}

units. If the new point

Q

lies in the third quadrant, then the equation of the line passing through

Q

and perpendicular to

L

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

If the orthocenter of the triangle formed by the lines

2 x + 3 y - 1 = 0, x + 2 y + 1 = 0

and

a x + b y - 1 = 0

lies at origin, then

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

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

Let

a, b

and

c

be distinct and none of them is equal to

1

. If the lines

x + a y + a = 0

b x + y + b = 0

and

c x + c y + 1 = 0

are concurrent, then the value of

\frac{a}{a - 1} + \frac{b}{b - 1} + \frac{c}{c - 1}

HARD

Mathematics>Coordinate Geometry>The Straight Line>Intersection of Straight Lines

If a variable line drawn through the intersection of the lines

\frac{x}{3} + \frac{y}{4} = 1

and

\frac{x}{4} + \frac{y}{3} = 1

, meets the coordinate axes at

A

and

B

(A \neq B),

then the locus of the midpoint of

AB

is:

If the lines 2x-3y+5=0,9x-5y+14=0 and 3x-7y+λ=0 are concurrent, then the value of λ is equal to

Important Questions on The Straight Line

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If the lines $2 x - 3 y + 5 = 0, 9 x - 5 y + 14 = 0$ and $3 x - 7 y + λ = 0$ are concurrent, then the value of $λ$ is equal to