Core Syllabus Module

Straight Lines

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Main 2022 (28 July Shift 1)

LEVELJEE Advanced

For $t \in (0, 2 π)$ , if $A B C$ is an equilateral triangle with vertices $A (sin t, cos t), B (cos t, sin t)$ and $C (a, b)$ such that its orthocentre lies on a circle with centre $(1, \frac{1}{3})$ , then $(a^{2} - b^{2})$ is equal to :

(A)

\frac{8}{3}

(B)

8

(C)

\frac{77}{9}

(D)

\frac{80}{9}

JEE Main 2025 (January)

LEVELJEE Advanced

Let $A (6, 8)$ , $B (10 cos α, - 10 sin α)$ and $C (- 10 sin α, 10 cos α)$ , be the vertices of a triangle. If $L (a, 9)$ and $G (h, k)$ be its orthocenter and centroid respectively, then $(5 a - 3 h + 6 k + 100 sin 2 α)$ is equal to

JEE Main 2021 (March)

LEVELJEE Advanced

Let $tan α, tan β$ and $tan γ; α, β, γ \neq = \frac{( 2 n - 1 ) π}{2}, n \in N$ be the slopes of three line segments OA OB and OC, respectively, where O is origin. If circumcentre of $△ A B C$ coincides with origin and its orthocentre lies on y -axis, then the value of $(\frac{c o s 3 α + c o s 3 β + c o s 3 γ}{c o s α c o s β c o s γ})^{2}$ is equal to :

JEE Main 2025 (January)

LEVELJEE Advanced

A rod of length eight units moves such that its ends A and B always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$ respectively. If the locus of the point P, that divides the rod AB internally in the ratio 2:1 is $9 (x^{2} + α y^{2} + β x y + γ x + 28 y) - 76 = 0$ , then $α - β - γ$ is equal to :

JEE Main 2022 (29 July Shift 2)

LEVELJEE Advanced

Let $m_{1}, m_{2}$ be the slopes of two adjacent sides of a square of side $a$ such that $a^{2} + 11 a + 3 (m_{1}^{2} + m_{2}^{2}) = 220$ . If one vertex of the square is $(10 (cos α - sin α), 10 (sin α + cos α))$ , where $α \in [0, π /2]$ and the equation of one diagonal is $(cos α - sin α) x + (sin α + cos α) y = 10$ , then $72 (sin^{4} α + cos^{4} α) + a^{2} - 3 a + 13$ is equal to:

(A)

119

(B)

128

(C)

145

(D)

155

JEE Main 2026 (28 January Shift 1)

LEVELJEE Advanced

Let $A B C$ be an equilateral triangle with orthocenter at the origin and the side $B C$ on the line $x + 22 y = 4$ . If the co-ordinates of the vertex A are $(α, β)$ , then the greatest integer less than or equal to $∣ α + 2 β ∣$ is

JEE Main 2022 (26 July Shift 1)

LEVELJEE Advanced

The equations of the sides $A B, B C$ and $C A$ of a triangle $A B C$ are $2 x + y = 0$ , $x + p y = 15 a$ and $x - y = 3$ respectively. If its orthocentre is $(2, a)$ , $- \frac{1}{2} < a < 2$ , then $p$ is equal to

JEE Main 2022 (28 June Shift 2)

LEVELJEE Advanced

Let a triangle be bounded by the lines $L_{1} : 2 x + 5 y = 10$ ; $L_{2} : - 4 x + 3 y = 12$ and the line $L_{3}$ , which passes through the point $P (2, 3)$ , intersect $L_{1}$ at $A$ and $L_{2}$ at $B$ . If the point $P$ divides the line-segment $A B$ , internally in the ratio $1 : 3$ , then the area of the triangle is equal to

(A)

\frac{110}{13}

(B)

\frac{132}{13}

(C)

\frac{142}{13}

(D)

\frac{151}{13}

JEE Main 2022 (28 June Shift 1)

LEVELJEE Advanced

A ray of light passing through the point $P (2, 3)$ reflects on the $x$ -axis at point $A$ and the reflected ray passes through the point $Q (5, 4)$ . Let $R$ be the point that divides the line segment $A Q$ internally into the ratio $2 : 1$ . Let the co-ordinates of the foot of the perpendicular $M$ from $R$ on the bisector of the angle $P A Q$ be $(α, β)$ . Then, the value of $7 α + 3 β$ is equal to _______.

JEE Main 2026 (21 January Shift 1)

LEVELJEE Advanced

Let a point A lie between the parallel lines $L_{1}$ and $L_{2}$ such that its distances from $L_{1}$ and $L_{2}$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle $A B C$ , where the points $B$ and $C$ lie on the lines $L_{1}$ and $L_{2}$ , respectively, is :

(A)

156

(B)

213

(C)

122

(D)

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 2026 (22 January Shift 2)

LEVELJEE Advanced

Among the statements (S1) : If $A (5, - 1)$ and $B (- 2, 3)$ are two vertices of a triangle, whose orthocentre is $(0, 0)$ , then its third vertex is $(- 4, - 7)$ and (S2) : If positive numbers 2a, b, c are three consecutive terms of an A.P., then the lines $a x + b y + c = 0$ are concurrent at $(2, - 2)$ ,

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 2023 (11 Apr Shift 2)

LEVELJEE Advanced

If the line $l_{1} : 3 y - 2 x = 3$ is the angular bisector of the lines $l_{2} : x - y + 1 = 0$ and $l_{3} : α x + β y + 17 = 0$ , then $α^{2} + β^{2} - α - β$ is equal to

JEE Main 2019 (9 January)

LEVELJEE Advanced

Let the equations of two sides of a triangle be $3 x - 2 y + 6 = 0$ and $4 x + 5 y - 20 = 0$ . If the orthocentre of this triangle is at (1,1), then the equation of its third side is :

(A)

122 y - 26 x - 1675 = 0

(B)

26 x + 61 y + 1675 = 0

(C)

122 y + 26 x + 1675 = 0

(D)

26 x - 122 y - 1675 = 0

JEE Main 2025 (January)

LEVELJEE Advanced

Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then $(QR)^{2}$ is equal to

JEE Main 2023 (24 January Shift 2)

LEVELJEE Advanced

The equations of the sides AB, BC and CA of a triangle ABC are: $2 x + y = 0$ , $x + p y = 21 a$ , $(a \neq = 0)$ and $x - y = 3$ respectively. Let $P (2, a)$ be the centroid of $Δ A B C$ . Then $(B C)^{2}$ is equal to

JEE Main 2023 (29 January Shift 1)

LEVELJEE Advanced

A light ray emits from the origin making an angle $3 0^{\circ}$ with the positive x-axis. After getting reflected by the line $x + y = 1$ , if this ray intersects x-axis at Q, then the abscissa of Q is

(A)

\frac{2}{3 - 1}

(B)

\frac{2}{3 + 3}

(C)

\frac{2}{3 - 3}

(D)

\frac{3}{2 ( 3 + 1 )}

JEE Advanced 1991

LEVELJEE Advanced

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 Main 2025 (January)

LEVELJEE Advanced

Let the line $x + y = 1$ meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is $\frac{4}{9}$ of the area of the triangle OAB and AN : NB = $λ$ : 1, then the sum of all possible value(s) of $λ$ is :

(A)

(B)

\frac{5}{2}

(C)

\frac{1}{2}

(D)

\frac{13}{6}

Micro-Topic Strategy Matrix

Distance and Section Formulas

Area of Triangle

Slope of a Line

Various Forms of Equations of a Line

Angle Between Two Lines

Distance of a Point from a Line

Family of Lines

Centroid, Incenter, Orthocenter, and Circumcenter

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equations of the sides AB,BC and CA of a triangle ABC are 2x+y=0, x+py=15a and x−y=3 respectively. If its orthocentre is (2,a), −21​<a<2, then p is equal to

.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; } If the line l1​:3y−2x=3 is the angular bisector of the lines l2​:x−y+1=0 and l3​:αx+βy+17=0, then α2+β2−α−β is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the equations of two sides of a triangle be 3x−2y+6=0 and 4x+5y−20=0. If the orthocentre of this triangle is at (1,1), then the equation of its third side is :

.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.

The equations of the sides $A B, B C$ and $C A$ of a triangle $A B C$ are $2 x + y = 0$ , $x + p y = 15 a$ and $x - y = 3$ respectively. If its orthocentre is $(2, a)$ , $- \frac{1}{2} < a < 2$ , then $p$ is equal to

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:

If the line $l_{1} : 3 y - 2 x = 3$ is the angular bisector of the lines $l_{2} : x - y + 1 = 0$ and $l_{3} : α x + β y + 17 = 0$ , then $α^{2} + β^{2} - α - β$ is equal to

Let the equations of two sides of a triangle be $3 x - 2 y + 6 = 0$ and $4 x + 5 y - 20 = 0$ . If the orthocentre of this triangle is at (1,1), then the equation of its third side is :

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 $y + 2 x = 5$ .