Two lines ( L 1 and L 2 ) are drawn from point α , α making an angle 45 ° with the lines L 3 ≡ x + y - f ( α ) = 0 and L 4 ≡ x + y + f ( α ) = 0 . L 1 intersects L 3 and L 4 at A and B and L 2 intersects L 3 and L 4 at C and D respectively ( | 2 α | > | f ( α ) | ) . If the area of trapezium A B D C is independent of α . If f ( α ) = λ α q , where λ is a constant, then | q | is

Given that equation of lines : L 3 : x + y - f α = 0 … 1 L 4 : x + y + f α = 0 … 2 Since the lines L 3 & L 4 are parallel, then distance between L 3 & L 4 = d = f α + f α 1 2 + 1 2 = 2 f α From diagram the equation of lines L 1 : y = α … 3 & L 2 : x = α … 4 So by solving equation 1 & 3 , we get coordinates of point A - α + f α , α Solving equation 2 & 3 , we get coordinates of point B - α - f α , α Solving equation 1 & 4 , we get coordinates of point C α , f α - α Solving equation 2 & 4 , we get coordinates of point D α , - α - f α Now, A C = 2 α - f α 2 + 2 α - f α 2 = 2 2 α - f α and B D = 2 α + f α 2 + - 2 α - f α 2 = 2 2 α + f α So, area of trapezium A B C D = 1 2 × A C + B D × d = 1 2 × 2 2 α - f α + 2 α + f α × 2 f α = f α 2 α - f α + 2 α + f α Since given that 2 α > f α ⇒ 2 α - f α > 0 Since 2 α - f α ≥ 2 α - f α > 0 ⇒ 2 α - f α > 0 ⇒ 2 α - f α = 2 α - f α Then area of trapezium A B C D = f α 2 α - f α + 2 α + f α = 4 α f α Since f α = λ α q Then area of trapezium = 4 α × λ α q = 4 λ α q + 1 Area of trapezium is independent of α , when q + 1 = 0 or q = - 1 Hence q = - 1 = 1

HARD

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IMPORTANT

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Drawn from the origin are two mutually perpendicular straight lines forming an isosceles triangle together with the straight line $2 x + y = 5 .$ Then, find the area of the triangle.

Important Questions on Point and Straight Line

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 25

On the straight-line

y = x + 2,

a point

(a, b)

is such that the sum of the square of distances from the straight lines

3 x - 4 y + 8 = 0

and

3 x - y - 1 = 0

is least, then find value of

11 (a + b) .

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 26

A

is a variable point on

x

-axis and

B (0, b)

is a fixed point. An equilateral triangle

A B C

is completed with vertex

C

away from origin. If the locus of the point

C

α x + β y = b,

then

α^{2} + β^{2}

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 27

Two lines (

L_{1}

and

L_{2}

) are drawn from point

(α, α)

making an angle

45 °

with the lines

L_{3} \equiv x + y - f (α) = 0

and

L_{4} \equiv x + y + f (α) = 0 . L_{1}

intersects

L_{3}

and

L_{4}

A

and

B

and

L_{2}

intersects

L_{3}

and

L_{4}

C

and

D

respectively

(| 2 α | > | f (α) |) .

If the area of trapezium

A B D C

is independent of

α

. If

f (α) = λ α^{q},

where

λ

is a constant, then

| q |

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 28

The portion of the line

a x + b y - 1 = 0,

intercepted between the lines

a x + y + 1 = 0

and

x + b y = 0

subtends a right-angle at the origin and the condition in

a

and

b

λ a + b + b^{2} = 0,

then find value of

λ

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 29

If the straight lines joining the origin and the points of intersection of the curve

5 x^{2} + 12 x y - 6 y^{2} + 4 x - 2 y + 3 = 0

and

x + k y - 1 = 0

are equally inclined to the

x

-axis, then find the value of

|k| .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 30

If the points of intersection of curves

C_{1} = 4 y^{2} - λ x^{2} - 2 x y - 9 x + 3

and

C_{2} = 2 x^{2} + 3 y^{2} - 4 x y + 3 x - 1

subtends a right angle at origin, then find the the value of

λ .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 31

A parallelogram is formed by the lines

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

and the lines through

(p, q)

parallel to them and the equation of the diagonal of the parallelogram which doesn't pass through origin is

(λ x - p) (a p + h q) + (μ y - q) (h p + b q) = 0,

then find the value of

λ^{3} + μ^{3} .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 32

The equation

9 x^{3} + 9 x^{2} y - 45 x^{2} = 4 y^{3} + 4 x y^{2} - 20 y^{2}

represents

3

straight lines, two of which passes through origin. Then find the area of the triangle formed by these lines

Drawn from the origin are two mutually perpendicular straight lines forming an isosceles triangle together with the straight line 2x+y=5 . Then, find the area of the triangle.

Important Questions on Point and Straight Line

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Drawn from the origin are two mutually perpendicular straight lines forming an isosceles triangle together with the straight line $2 x + y = 5 .$ Then, find the area of the triangle.