A line ' L ' is drawn from P 4 , 3 to meet the lines L 1 : 3 x + 4 y + 5 = 0 and L 2 : 3 x + 4 y + 15 = 0 at point A and B respectively. From ' A ' a line, perpendicular to L is drawn meeting the line L 2 at A 1 . Similarly from point ' B ' a line, perpendicular to L is drawn meeting the line L 1 at B 1 . Thus a parallelogram A A 1 B B 1 is formed. Find the equation(s) of ' L ' so that the area of the parallelogram A A 1 B B 1 is least.

Given: A line L is drawn from point P 4 , 3 meet to the two lines 3 x + 4 y + 5 = 0 and 3 x + 4 y + 15 = 0 at A and B . Both the given lines L 1 and L 2 are parallel. Distance between the parallel lines is = c 2 - c 1 a 2 + b 2 So, the distance between these lines is, ⇒ 15 - 5 3 2 + 4 2 ⇒ 2 units. Let ∠ B C A = θ . So, A B = B C cosec θ = 2 cosec θ And A A 1 = A D sec θ = 2 sec θ Now, the area of parallelogram A A 1 B B 1 is, ⇒ A B × A A 1 = 2 cosec θ × 2 sec θ ⇒ A B × A A 1 = 4 sec θ cosec θ ⇒ A B × A A 1 = 4 sin θ cos θ ⇒ A B × A A 1 = 8 sin 2 θ . So, the area of parallelogram is least for angle π 4 . Let the slope of the line is m. So, from slope formula we have, 1 = m + 3 4 1 - 3 m 4 ⇒ 1 = 4 m + 3 4 - 3 m ⇒ 4 m + 3 = ± 4 - 3 m ⇒ m = 1 7 , - 7 So, form two point slope form we get the equation of line, y - 3 = 1 7 x - 4 or y - 3 = - 7 x - 4 x - 7 y + 17 = 0 or 7 x + y - 31 = 0 .

HARD

JEE Main/Advanced

IMPORTANT

Earn 100

If the points $(4, 7)$ and $(\cos θ, \sin θ)$ , where $0 < θ < π$ , lie on the same side of the line $x + y - 1 = 0$ , then prove that $θ$ lies in the first quadrant.

Important Questions on The Straight Lines

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 2 - The Straight Lines>INTRODUCTORY EXERCISE 2.2>Q 3

Find the equations of lines parallel to

3 x - 4 y - 5 = 0

at a unit distance from it.

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 2 - The Straight Lines>INTRODUCTORY EXERCISE 2.2>Q 4

Prove that the area of the parallelogram formed by the lines

3 x - 4 y + a = 0, 3 x - 4 y + 3 a = 0, 4 x - 3 y - a = 0

and

4 x - 3 y - 2 a = 0

\frac{2 a^{2}}{7}

sq. units

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 2 - The Straight Lines>INTRODUCTORY EXERCISE 2.2>Q 5

A line '

L

' is drawn from

P (4, 3)

to meet the lines

L_{1} : 3 x + 4 y + 5 = 0

and

L_{2} : 3 x + 4 y + 15 = 0

at point

A

and

B

respectively. From '

A

' a line, perpendicular to

L

is drawn meeting the line

L_{2}

A_{1} .

Similarly from point '

B

' a line, perpendicular to

L

is drawn meeting the line

L_{1}

B_{1} .

Thus a parallelogram

A A_{1} B B_{1}

is formed. Find the equation(s) of '

L

' so that the area of the parallelogram

A A_{1} B B_{1}

is least.

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 2 - The Straight Lines>INTRODUCTORY EXERCISE 2.2>Q 6

The vertices of a triangle

O B C

are

O (0, 0) B (- 3, - 1)

and

C (- 1, - 3) .

Find the equation of line parallel to

B C

and intersecting the sides

O B

and

O C,

whose perpendicular distance from the point

(0, 0)

\frac{1}{2}

units.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 2 - The Straight Lines>INTRODUCTORY EXERCISE 2.3>Q 2

a, b, c

are in

AP

then

a x + b y + c = 0

represents :

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 2 - The Straight Lines>INTRODUCTORY EXERCISE 2.3>Q 3

If the lines

x + 2 a y + a = 0, x + 3 b y + b = 0

and

x + 4 c y + c = 0

are concurrent, then

a, b

and

c

are in

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 2 - The Straight Lines>INTRODUCTORY EXERCISE 2.3>Q 4

The line

a x + b y + c = 0

, where

3 a + 2 b + 4 c = 0

is concurrent at the point

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 2 - The Straight Lines>INTRODUCTORY EXERCISE 2.3>Q 6

The straight line

x + 2 y - 9 = 0, 3 x + 5 y - 5 = 0

and

a x + b y - 1 = 0

are concurrent if the straight line

35 x - 22 y + 1 = 0

passes through the point

If the points 4,7 and cosθ,sinθ, where 0<θ<π, lie on the same side of the line x+y-1=0, then prove that θ lies in the first quadrant.

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

If the points $(4, 7)$ and $(\cos θ, \sin θ)$ , where $0 < θ < π$ , lie on the same side of the line $x + y - 1 = 0$ , then prove that $θ$ lies in the first quadrant.