LFM Pure

jee-main 2013 Q68 Reflection and Image in a Line View

If the image of point $\mathrm { P } ( 2,3 )$ in a line L is $\mathrm { Q } ( 4,5 )$, then the image of point $\mathrm { R } ( 0,0 )$ in the same line is:
(1) $( 2,2 )$
(2) $( 4,5 )$
(3) $( 3,4 )$
(4) $( 7,7 )$

jee-main 2013 Q69 Triangle Properties and Special Points View

If the three lines $x - 3 y = p , a x + 2 y = q$ and $a x + y = r$ form a right-angled triangle then :
(1) $a ^ { 2 } - 9 a + 18 = 0$
(2) $a ^ { 2 } - 6 a - 12 = 0$
(3) $a ^ { 2 } - 6 a - 18 = 0$
(4) $a ^ { 2 } - 9 a + 12 = 0$

jee-main 2013 Q69 Reflection and Image in a Line View

A ray of light along $x + \sqrt{3}y = \sqrt{3}$ gets reflected upon reaching $X$-axis, the equation of the reflected ray is
(1) $y = \sqrt{3}x - \sqrt{3}$
(2) $\sqrt{3}y = x - 1$
(3) $y = x + \sqrt{3}$
(4) $\sqrt{3}y = x - \sqrt{3}$

jee-main 2013 Q70 Triangle Properties and Special Points View

The $x$-coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as $(0,1)$, $(1,1)$ and $(1,0)$ is
(1) $1 + \sqrt{2}$
(2) $1 - \sqrt{2}$
(3) $2 + \sqrt{2}$
(4) $2 - \sqrt{2}$

jee-main 2013 Q74 Area Computation in Coordinate Geometry View

If the extremities of the base of an isosceles triangle are the points $( 2 a , 0 )$ and $( 0 , a )$ and the equation of one of the sides is $x = 2 a$, then the area of the triangle, in square units, is :
(1) $\frac { 5 } { 4 } a ^ { 2 }$
(2) $\frac { 5 } { 2 } a ^ { 2 }$
(3) $\frac { 25 a ^ { 2 } } { 4 }$
(4) $5 a^2$

jee-main 2014 Q67 Point-to-Line Distance Computation View

If a line $L$ is perpendicular to the line $5 x - y = 1$, and the area of the triangle formed by the line $L$ and the coordinate axes is 5 sq units, then the distance of the line $L$ from the line $x + 5 y = 0$ is
(1) $\frac { 7 } { \sqrt { 13 } }$ units
(2) $\frac { 7 } { \sqrt { 5 } }$ units
(3) $\frac { 5 } { \sqrt { 13 } }$ units
(4) $\frac { 5 } { \sqrt { 7 } }$ units

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) $x - y + 7 = 0$

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144