LFM Pure

jee-main 2014 Q68 Line Equation and Parametric Representation View

Let $P S$ be the median of the triangle with vertices $P ( 2,2 ) , Q ( 6 , - 1 )$ and $R ( 7,3 )$. The equation of the line passing through $( 1 , - 1 )$ and parallel to $P S$ is
(1) $4 x + 7 y + 3 = 0$
(2) $2 x - 9 y - 11 = 0$
(3) $4 x - 7 y - 11 = 0$
(4) $2 x + 9 y + 7 = 0$

jee-main 2014 Q68 Triangle Properties and Special Points View

The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points $\left( a ^ { 2 } + 1 , a ^ { 2 } + 1 \right)$ and $( 2 a , - 2 a ) , a \neq 0$. Then for any $a$, the orthocentre of this triangle lies on the line
(1) $y - \left( a ^ { 2 } + 1 \right) x = 0$
(2) $y - 2 a x = 0$
(3) $y + x = 0$
(4) $( a - 1 ) ^ { 2 } x - ( a + 1 ) ^ { 2 } y = 0$

jee-main 2014 Q69 Collinearity and Concurrency View

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
(1) $3 b c - 2 a d = 0$
(2) $3 b c + 2 a d = 0$
(3) $2 b c - 3 a d = 0$
(4) $2 b c + 3 a d = 0$

jee-main 2014 Q70 Triangle Properties and Special Points View

Given three points $P , Q , R$ with $P ( 5,3 )$ and $R$ lies on the $x$-axis. If the equation of $RQ$ is $x - 2 y = 2$ and $PQ$ is parallel to the $x$-axis, then the centroid of $\triangle PQR$ lies on the line
(1) $x - 2 y + 1 = 0$
(2) $2 x + y - 9 = 0$
(3) $2 x - 5 y = 0$
(4) $5 x - 2 y = 0$

jee-main 2015 Q69 Locus Determination View

Locus of the image of the point $( 2,3 )$ in the line $( 2 x - 3 y + 4 ) + k ( x - 2 y + 3 ) = 0 , k \in \mathbb{R}$, is a
(1) Circle of radius $\sqrt { 3 }$
(2) Straight line parallel to $x$-axis.
(3) Straight line parallel to $y$-axis.
(4) Circle of radius $\sqrt { 2 }$

jee-main 2016 Q69 Section Ratio and Division of Segments View

A straight line through origin $O$ meets the lines $3y = 10 - 4x$ and $8x + 6y + 5 = 0$ at points $A$ and $B$ respectively. Then $O$ divides the segment $AB$ in the ratio: (1) $2:3$ (2) $1:2$ (3) $4:1$ (4) $3:4$

jee-main 2016 Q69 Section Ratio and Division of Segments View

A straight line through origin $O$ meets the lines $3 y = 10 - 4 x$ and $8 x + 6 y + 5 = 0$ at points $A$ and $B$ respectively. Then, $O$ divides the segment $A B$ in the ratio
(1) $2 : 3$
(2) $1 : 2$
(3) $4 : 1$
(4) $3 : 4$

jee-main 2016 Q70 Reflection and Image in a Line View

A ray of light is incident along a line which meets another line $7 x - y + 1 = 0$ at the point $( 0,1 )$. The ray is then reflected from this point along the line $y + 2 x = 1$. Then the equation of the line of incidence of the ray of light is :
(1) $41 x - 25 y + 25 = 0$
(2) $41 x + 25 y - 25 = 0$
(3) $41 x - 38 y + 38 = 0$
(4) $41 x + 38 y - 38 = 0$

jee-main 2016 Q72 Geometric Figure on Coordinate Plane View

Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7x - y - 5 = 0$. If its diagonals intersect at $(-1, -2)$, then which one of the following is a vertex of this rhombus?
(1) $(-3, -9)$
(2) $(-3, -8)$
(3) $\left(\frac{1}{3}, -\frac{8}{3}\right)$
(4) $\left(-\frac{1}{3}, -\frac{8}{3}\right)$

jee-main 2017 Q67 Triangle Properties and Special Points View

Let $k$ be an integer such that the triangle with vertices $(k, -3)$, $(5, k)$ and $(-k, 2)$ has area 28 sq. units. Then the orthocenter of this triangle is at the point:
(1) $\left(2, -\dfrac{1}{2}\right)$
(2) $\left(1, \dfrac{3}{4}\right)$
(3) $\left(1, -\dfrac{3}{4}\right)$
(4) $\left(2, \dfrac{1}{2}\right)$

jee-main 2017 Q75 Locus Determination View

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:
(1) $7 x y = 6 ( x + y )$
(2) $4 ( x + y ) ^ { 2 } - 28 ( x + y ) + 49 = 0$
(3) $6 x y = 7 ( x + y )$
(4) $14 ( x + y ) ^ { 2 } - 97 ( x + y ) + 168 = 0$

jee-main 2018 Q68 Locus Determination View

A straight line through a fixed point $( 2,3 )$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $O P R Q$ is completed, then the locus of $R$ is:
(1) $3 x + 2 y = 6 x y$
(2) $3 x + 2 y = 6$
(3) $2 x + 3 y = x y$
(4) $3 x + 2 y = x y$

jee-main 2018 Q68 Area Computation in Coordinate Geometry View

In a triangle $A B C$, coordiantates of $A$ are $( 1,2 )$ and the equations of the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$ respectively. Then area of $\triangle A B C$ (in sq. units) is
(1) 5
(2) 9
(3) 12
(4) 4

jee-main 2018 Q75 Area Computation in Coordinate Geometry View

In a triangle $ABC$, coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are respectively, $x + y = 5$ and $x = 4$. Then area of $\triangle ABC$ (in sq. units) is :
(1) 12
(2) 4
(3) 9
(4) 5

jee-main 2018 Q77 Triangle Properties and Special Points View

Let the orthocentre and centroid of a triangle be $A ( - 3,5 )$ and $B ( 3,3 )$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $A C$ as diameter, is:
(1) $\frac { 3 \sqrt { } 5 } { 2 }$
(2) $\sqrt { 10 }$
(3) $2 \sqrt { 10 }$
(4) $3 \sqrt { \frac { 5 } { 2 } }$

jee-main 2019 Q66 Triangle Properties and Special Points View

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?

jee-main 2019 Q67 Slope and Angle Between Lines View

Suppose that the points $( h , k )$, $( 1,2 )$ and $( - 3,4 )$ lie on the line $L _ { 1 }$. If a line $L _ { 2 }$ passing through the points $( h , k )$ and $( 4,3 )$ is perpendicular to $L _ { 1 }$, then $\frac { k } { h }$ equals:
(1) $- \frac { 1 } { 7 }$
(2) 3
(3) 0
(4) $\frac { 1 } { 3 }$

jee-main 2019 Q68 Triangle Properties and Special Points View

If the line $3 x + 4 y - 24 = 0$ intersects the $x$-axis is at the point $A$ and the $y$-axis at the point $B$, then the incentre of the triangle $O A B$, where $O$ is the origin, is:
(1) $( 4,4 )$
(2) $( 3,4 )$
(3) $( 4,3 )$
(4) $( 2,2 )$

jee-main 2019 Q68 Line Equation and Parametric Representation View

If a straight line passing through the point $P ( - 3,4 )$ is such that its intercepted portion between the coordinate axes is bisected at $P$, then its equation is :
(1) $4 x + 3 y = 0$
(2) $4 x - 3 y + 24 = 0$
(3) $3 x - 4 y + 25 = 0$
(4) $x - y + 7 = 0$

jee-main 2019 Q68 Point-to-Line Distance Computation View

A point on the straight line, $3x + 5y = 15$ which is equidistant from the coordinate axes will lie only in:
(1) $1^{\text{st}}$ and $2^{\text{nd}}$ quadrants
(2) $1^{\text{st}}$, $2^{\text{nd}}$ and $4^{\text{th}}$ quadrants
(3) $1^{\text{st}}$ quadrant
(4) $4^{\text{th}}$ quadrant

jee-main 2019 Q68 Slope and Angle Between Lines View

Slope of a line passing through $P ( 2,3 )$ and intersecting the line $x + y = 7$ at a distance of 4 units from $P$, is
(1) $\frac { \sqrt { 7 } - 1 } { \sqrt { 7 } + 1 }$
(2) $\frac { 1 - \sqrt { 7 } } { 1 + \sqrt { 7 } }$
(3) $\frac { \sqrt { 5 } - 1 } { \sqrt { 5 } + 1 }$
(4) $\frac { 1 - \sqrt { 5 } } { 1 + \sqrt { 5 } }$

jee-main 2019 Q68 Point-to-Line Distance Computation View

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
(1) $\frac { 2 } { \sqrt { 5 } }$
(2) $\frac { \sqrt { 2 } } { 5 }$
(3) $\frac { 2 } { 5 }$
(4) $\sqrt { \frac { 2 } { 5 } }$

jee-main 2019 Q69 Locus Determination View

A point $P$ moves on the line $2 x - 3 y + 4 = 0$. If $Q ( 1,4 )$ and $R ( 3 , - 2 )$ are fixed points, then the locus of the centroid of $\triangle P Q R$ is a line:
(1) with slope $\frac { 2 } { 3 }$
(2) with slope $\frac { 3 } { 2 }$
(3) parallel to $y$-axis
(4) parallel to $x$-axis

jee-main 2019 Q70 Triangle Properties and Special Points View

Let the equations of two sides of a triangle be $3x - 2y + 6 = 0$ and $4x + 5y - 20 = 0$. If the orthocenter of this triangle is at $(1,1)$ then the equation of its third side is:
(1) $122y + 26x + 1675 = 0$
(2) $26x - 122y - 1675 = 0$
(3) $26x + 61y + 1675 = 0$
(4) $122y - 26x - 1675 = 0$

jee-main 2019 Q75 Perspective, Projection, and Applied Geometry View

Two vertical poles of height, $20 m$ and $80 m$ stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is:
(1) 16
(2) 12
(3) 18
(4) 15

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164