Angle Between Two Lines

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JEE Main 2011

LEVELJEE Main

The lines $L_{1} : y - x = 0$ and $L_{2} : 2 x + y = 0$ intersect the line $L_{3} : y + 2 = 0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$ . Statement-1: The ratio $P R : R Q$ equals $22 : 5$ . Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

JEE Main 2009

LEVELJEE Main

The lines $p (p^{2} + 1) x - y + q = 0$ and $(p^{2} + 1)^{2} x + (p^{2} + 1) y + 2 q = 0$ are perpendicular to a common line for

(A)

exactly one values of

p

(B)

exactly two values of

p

(C)

more than two values of

p

(D)

no value of

p

JEE Main 2007

LEVELJEE Main

If one of the lines of $m y^{2} + (1 - m^{2}) x y - m x^{2} = 0$ is a bisector of the angle between the lines $x y = 0$ , then $m$ is

If the pair of lines $a x^{2} + 2 (a + b) x y + b y^{2} = 0$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then

(A)

3 a^{2} - 10 ab + 3 b^{2} = 0

(B)

3 a^{2} - 2 ab + 3 b^{2} = 0

(C)

3 a^{2} + 10 ab + 3 b^{2} = 0

(D)

3 a^{2} + 2 ab + 3 b^{2} = 0

JEE Main 2003

LEVELJEE Main

If the pair of straight lines $x^{2} - 2 p x y - y^{2} = 0$ and $x^{2} - 2 q x y - y^{2} = 0$ be such that each pair bisects the angle between the other pair, then

(A)

pq = - 1

(B)

p = q

(C)

p = - q

(D)

pq = 1

JEE Main 2002

LEVELBoard

The pair of lines represented by $3 a x^{2} + 5 x y + (a^{2} - 2) y^{2} = 0$ are perpendicular to each other for

(A)

two values of

a

(B)

\forall a

(C)

for one value of

a

(D)

for no values of

a

JEE Advanced 2011

LEVELJEE Main

A straight line $L$ through the point $(3, - 2)$ is inclined at an angle $6 0^{\circ}$ to the line $3 x + y = 1$ . If $L$ also intersects the x-axis, then the equation of $L$ is

(A)

y + 3 x + 2 - 33 = 0

(B)

y - 3 x + 2 + 33 = 0

(C)

3 y - x + 3 + 23 = 0

(D)

3 y + x - 3 + 23 = 0

JEE Advanced 2007

LEVELJEE Main

Lines $L_{1} : y - x = 0$ and $L_{2} : 2 x + y = 0$ intersect the line $L_{3} : y + 2 = 0$ at $P$ and $Q$ , respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$ . STATEMENT-1 : The ratio $P R : R Q$ equals $22 : 5$ . because STATEMENT-2 : In any triangle, bisector of an angle divides the triangle into two similar triangles.

(A)

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

(B)

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

(C)

Statement-1 is True, Statement-2 is False

(D)

Statement-1 is False, Statement-2 is True

JEE Advanced 2004

LEVELJEE Main

Area of the triangle formed by the line $x + y = 3$ and angle bisectors of the pair of straight lines $x^{2} - y^{2} + 2 y = 1$ is

Let $P QR$ be a right angled isosceles triangle, right angled at $P (2, 1)$ . If the equation of the line $QR$ is $2 x + y = 3$ , then the equation representing the pair of lines $P Q$ and $P R$ is

(A)

3 x^{2} - 3 y^{2} + 8 x y + 20 x + 10 y + 25 = 0

(B)

3 x^{2} - 3 y^{2} + 8 x y - 20 x - 10 y + 25 = 0

(C)

3 x^{2} - 3 y^{2} + 8 x y + 10 x + 15 y + 20 = 0

(D)

3 x^{2} - 3 y^{2} - 8 x y - 10 x - 15 y - 20 = 0

JEE Advanced 1998

LEVELBoard

The diagonals of a parallelogram $P QR S$ are along the lines $x + 3 y = 4$ and $6 x - 2 y = 7$ . Then $P QR S$ must be a

Find the equation of the line passing through the point $(2, 3)$ and making intercept of length 2 units between the lines $y + 2 x = 3$ and $y + 2 x = 5$ .

JEE Advanced 1990

LEVELJEE Main

Straight lines $3 x + 4 y = 5$ and $4 x - 3 y = 15$ intersect at the point $A$ . Points $B$ and $C$ are chosen on these two lines such that $A B = A C$ . Determine the possible equations of the line $B C$ passing through the point $(1, 2)$ .

JEE Advanced 1985

LEVELJEE Main

Two sides of a rhombus $A B C D$ are parallel to the lines $y = x + 2$ and $y = 7 x + 3$ . If the diagonals of the rhombus intersect at the point $(1, 2)$ and the vertex $A$ is on the y-axis, find possible co-ordinates of $A$ .

JEE Advanced 1984

LEVELJEE Main

Two equal sides of an isosceles triangle are given by the equations $7 x - y + 3 = 0$ and $x + y - 3 = 0$ and its third side passes through the point $(1, - 10)$ . Determine the equation of the third side.

JEE Advanced 1989

LEVELJEE Advanced

Let $A B C$ be a triangle with $A B = A C$ . If $D$ is the midpoint of $B C$ , $E$ is the foot of the perpendicular drawn from $D$ to $A C$ and $F$ the mid-point of $D E$ , prove that $A F$ is perpendicular to $B E$ .

JEE Main 2018 (15 April Evening)

LEVELJEE Main

The sides of a rhombus ABCD are parallel to the lines, $x - y + 2 = 0$ and $7 x - y + 3 = 0$ . If the diagonals of the rhombus intersect at P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the ordinate of A is :-

JEE Main 2019 (12 January Shift 1)

LEVELBoard

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 :-

(A)

-5

(B)

- 35/3

(C)

35/3

(D)

JEE Main 2019 (9 April)

LEVELJEE Main

If the two lines $x + (a - 1) y = 1$ and $2 x + a^{2} y = 1 (a \in R - {0, 1})$ are perpendicular, then the distance of their point of intersection from the origin is :-

(A)

\frac{2}{5}

(B)

\frac{2}{5}

(C)

\frac{2}{5}

(D)

\frac{2}{5}

JEE Main 2021 (27 Aug Shift 2)

LEVELJEE Main

Two poles, AB of length $a$ metres and CD of length $a + b (b \neq = a)$ metres are erected at the same horizontal level with bases at B and D. If $B D = x$ and $tan ∠ A C B = \frac{1}{2}$ , then:

(A)

x^{2} + 2 (a + 2 b) x - b (a + b) = 0

(B)

x^{2} + 2 (a + 2 b) x + a (a + b) = 0

(C)

x^{2} - 2 a x + b (a + b) = 0

(D)

x^{2} - 2 a x + a (a + b) = 0

JEE Main 2021 (25 July Shift 2)

LEVELJEE Main

Let the equation of the pair of lines, $y = p x$ and $y = q x$ , can be written as $(y - p x) (y - q x) = 0$ . Then the equation of the pair of the angle bisectors of the lines $x^{2} - 4 x y - 5 y^{2} = 0$ is:

(A)

x^{2} - 3 x y + y^{2} = 0

(B)

x^{2} + 4 x y - y^{2} = 0

(C)

x^{2} + 3 x y - y^{2} = 0

(D)

x^{2} - 3 x y - y^{2} = 0

JEE Main 2021 (26 Aug Shift 1)

LEVELJEE Advanced

Let $A B C$ be a triangle with $A (- 3, 1)$ and $∠ A C B = θ, 0 < θ < \frac{π}{2}$ . If the equation of the median through $B$ is $2 x + y - 3 = 0$ and the equation of angle bisector of $C$ is $7 x - 4 y - 1 = 0$ , then $tan θ$ is equal to:

(A)

\frac{1}{2}

(B)

\frac{3}{4}

(C)

\frac{4}{3}

(D)

2

JEE Main 2021 (February) (25 February Shift 2)

LEVELJEE Advanced

If the curve $x^{2} + 2 y^{2} = 2$ intersects the line $x + y = 1$ at two points $P$ and $Q$ , then the angle subtended by the line segment $P Q$ at the origin is:

(A)

\frac{π}{2} + tan^{- 1} (\frac{1}{4})

(B)

\frac{π}{2} - tan^{- 1} (\frac{1}{4})

(C)

\frac{π}{2} + tan^{- 1} (\frac{1}{2})

(D)

\frac{π}{2} - tan^{- 1} (\frac{1}{2})

JEE Main 2021 (March)

LEVELJEE Main

The equation of one of the straight lines which passes through the point (1,3) and makes an angles $tan^{- 1} (2)$ with the straight line, $y + 1 = 32 x$ is

(A)

42 x + 5 y - (15 + 42) = 0

(B)

52 x + 4 y - (15 + 42) = 0

(C)

42 x + 5 y - 42 = 0

(D)

42 x - 5 y - (5 + 42) = 0

JEE Main 2022 (29 June Shift 1)

LEVELJEE Main

The distance between the two points $A$ and $A^{'}$ which lie on $y = 2$ such that both the line segments $A B$ and $A^{'} B$ (where $B$ is the point $(2, 3)$ ) subtend angle $\frac{π}{4}$ at the origin, is equal to :

(A)

(B)

\frac{48}{5}

(C)

\frac{52}{5}

(D)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The lines p(p2+1)x−y+q=0 and (p2+1)2x+(p2+1)y+2q=0 are perpendicular to a common line for

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one of the lines of my2+(1−m2)xy−mx2=0 is a bisector of the angle between the lines xy=0, then m is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the pair of straight lines x2−2pxy−y2=0 and x2−2qxy−y2=0 be such that each pair bisects the angle between the other pair, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The pair of lines represented by 3ax2+5xy+(a2−2)y2=0 are perpendicular to each other for

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A straight line L through the point (3,−2) is inclined at an angle 60∘ to the line 3​x+y=1. If L also intersects the x-axis, then the equation of L is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Area of the triangle formed by the line x+y=3 and angle bisectors of the pair of straight lines x2−y2+2y=1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let PQR be a right angled isosceles triangle, right angled at P(2,1). If the equation of the line QR is 2x+y=3, then the equation representing the pair of lines PQ and PR is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The diagonals of a parallelogram PQRS are along the lines x+3y=4 and 6x−2y=7. Then PQRS must be a

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the equation of the line passing through the point (2,3) and making intercept of length 2 units between the lines y+2x=3 and y+2x=5.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the straight line, 2x−3y+17=0 is perpendicular to the line passing through the points (7,17) and (15,β), then β equals :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the two lines x+(a−1)y=1 and 2x+a2y=1(a∈R−{0,1}) are perpendicular, then the distance of their point of intersection from the origin is :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two poles, AB of length a metres and CD of length a+b(b=a) metres are erected at the same horizontal level with bases at B and D. If BD=x and tan∠ACB=21​, then:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the curve x2+2y2=2 intersects the line x+y=1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of one of the straight lines which passes through the point (1,3) and makes an angles tan−1(2​) with the straight line, y+1=32​x is

The lines $p (p^{2} + 1) x - y + q = 0$ and $(p^{2} + 1)^{2} x + (p^{2} + 1) y + 2 q = 0$ are perpendicular to a common line for

If one of the lines of $m y^{2} + (1 - m^{2}) x y - m x^{2} = 0$ is a bisector of the angle between the lines $x y = 0$ , then $m$ is

If the pair of straight lines $x^{2} - 2 p x y - y^{2} = 0$ and $x^{2} - 2 q x y - y^{2} = 0$ be such that each pair bisects the angle between the other pair, then

The pair of lines represented by $3 a x^{2} + 5 x y + (a^{2} - 2) y^{2} = 0$ are perpendicular to each other for

A straight line $L$ through the point $(3, - 2)$ is inclined at an angle $6 0^{\circ}$ to the line $3 x + y = 1$ . If $L$ also intersects the x-axis, then the equation of $L$ is

Area of the triangle formed by the line $x + y = 3$ and angle bisectors of the pair of straight lines $x^{2} - y^{2} + 2 y = 1$ is

Let $P QR$ be a right angled isosceles triangle, right angled at $P (2, 1)$ . If the equation of the line $QR$ is $2 x + y = 3$ , then the equation representing the pair of lines $P Q$ and $P R$ is

The diagonals of a parallelogram $P QR S$ are along the lines $x + 3 y = 4$ and $6 x - 2 y = 7$ . Then $P QR S$ must be a

Find the equation of the line passing through the point $(2, 3)$ and making intercept of length 2 units between the lines $y + 2 x = 3$ and $y + 2 x = 5$ .

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 :-

If the two lines $x + (a - 1) y = 1$ and $2 x + a^{2} y = 1 (a \in R - {0, 1})$ are perpendicular, then the distance of their point of intersection from the origin is :-

Two poles, AB of length $a$ metres and CD of length $a + b (b \neq = a)$ metres are erected at the same horizontal level with bases at B and D. If $B D = x$ and $tan ∠ A C B = \frac{1}{2}$ , then:

If the curve $x^{2} + 2 y^{2} = 2$ intersects the line $x + y = 1$ at two points $P$ and $Q$ , then the angle subtended by the line segment $P Q$ at the origin is:

The equation of one of the straight lines which passes through the point (1,3) and makes an angles $tan^{- 1} (2)$ with the straight line, $y + 1 = 32 x$ is