Straight Lines · JEE Main previous year questions

Let $A(-2,-1),B(1,0),C(\alpha ,\beta )$ and $D(\gamma ,\delta )$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2x-y=5$ and the point $D$ lies on $3x-2y=6$, then the value of $|\alpha +\beta +\gamma +\delta |$ is equal to ____.

A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a,b,c)$, where $a,b,c$ are non-negative integers, on the line $AB$ lies at a distance of 21 units, from the point $A$. The distance between the points $P(a,b,c)$ and $Q(4,-12,3)$ is equal to ____.

In a $△ABC$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC$ is $2x-y=2$. If $2AB=BC$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha ,\beta )$, then $\alpha +2\beta$ is equal to

1. A.$42$
2. B.$39$
3. C.$48$
4. D.$45$

Let $PQR$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the mid point of PQ. The distance of the centroid of $△\text{PQR}$ from the point of intersection of the lines $\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}$ and $\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1}$ is

1. A.$69$
2. B.$\sqrt{99}$
3. C.$\sqrt{69}$
4. D.$9$

Let A be the point of intersection of the lines $3x+2y=14,5x-y=6$ and B be the point of intersection of the lines $4x+3y=8,6x+y=5$. The distance of the point $P(5,-2)$ from the line AB is

1. A.$\frac{13}{2}$
2. B.8
3. C.$\frac{5}{2}$
4. D.6

The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt{3}x-y+1=0$, is equal to

1. A.$3+4\sqrt{2}$
2. B.$6\sqrt{3}$
3. C.$4+6\sqrt{3}$
4. D.$4\sqrt{2}$

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$, and ${\mathrm{I}}_1=\underset{a}{\overset{b}{\int }}x\sin \left(4x-x^2\right)\mathrm{d}x$, ${\mathrm{I}}_2=\underset{a}{\overset{b}{\int }}\sin \left(4x-x^2\right)\mathrm{d}x$, then $36\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}$ is equal to :

1. A.80
2. B.72
3. C.66
4. D.88

Let $(\alpha ,\beta )$ be the centroid of the triangle formed by the lines $15x-y=82$, $6x-5y=-4$ and $9x+4y=17$. Then $\alpha +2\beta$ and $2\alpha -\beta$ are the roots of the equation :

1. A.$x^2-7x+12=0$
2. B.$x^2-13x+42=0$
3. C.$x^2-14x+48=0$
4. D.$x^2-10x+25=0$

If the line $l_1:3y-2x=3$ is the angular bisector of the lines $l_2:x-y+1=0$ and $l_3:\alpha x+\beta y+17=0$, then ${\alpha }^2+{\beta }^2-\alpha -\beta$ is equal to _________.

Let $C(\alpha ,\beta )$ be the circumcenter of the triangle formed by the lines $4x+3y=69$ $4y-3x=17$, and $x+7y=61$. Then $(\alpha -\beta {)}^2+\alpha +\beta$ is equal to :

1. A.$15$
2. B.$17$
3. C.$16$
4. D.$18$

The straight lines ${\text{l}}_1$ and ${\text{l}}_2$ pass through the origin and trisect the line segment of the line L : $9x+5y=45$ between the axes. If ${\text{m}}_1$ and ${\text{m}}_2$ are the slopes of the lines ${\text{l}}_1$ and ${\text{l}}_2$, then the point of intersection of the line $y=({\text{m}}_1+{\text{m}}_2)x$ with L lies on :

1. A.$6x-y=15$
2. B.$6x+y=10$
3. C.$y-x=5$
4. D.$y-2x=5$

For all $z\in \mathbb{C}$ on the curve $C_1:|z|=4$, let the locus of the point $z+\frac{1}{z}$ be the curve $C_2$. Then :

1. A.the curves $C_1$ and $C_2$ intersect at 4 points
2. B.the curve $C_2$ lies inside $C_1$
3. C.the curve $C_1$ lies inside $C_2$
4. D.the curves $C_1$ and $C_2$ intersect at 2 points

A straight line cuts off the intercepts $\mathrm{OA}=a$ and $\mathrm{OB}=b$ on the positive directions of $x$-axis and $y$ axis respectively. If the perpendicular from origin $O$ to this line makes an angle of $\frac{\pi }{6}$ with positive direction of $y$-axis and the area of $△\mathrm{OAB}$ is $\frac{98}{3}\sqrt{3}$, then $a^2-b^2$ is equal to :

1. A.$\frac{392}{3}$
2. B.98
3. C.196
4. D.$\frac{196}{3}$

A light ray emits from the origin making an angle $30^{\circ }$ 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 :

1. A.$\frac{2}{(\sqrt{3}-1)}$
2. B.$\frac{2}{3-\sqrt{3}}$
3. C.$\frac{\sqrt{3}}{2(\sqrt{3}+1)}$
4. D.$\frac{2}{3+\sqrt{3}}$

The equations of the sides AB, BC and CA of a triangle ABC are : $2x+y=0,x+py=21a,(a\ne 0)$ and $x-y=3$ respectively. Let P(2, a) be the centroid of $\Delta \mathrm{ABC}$. Then $(\mathrm{BC}{)}^2$ is equal to _______.

Let the circumcentre of a triangle with vertices $\mathrm{A}(a,3)$, $\mathrm{B}(b,5)$ and $\mathrm{C}(a,b)$, $ab>0$ be $\mathrm{P}(1,1)$. If the line $\mathrm{AP}$ intersects the line $\mathrm{BC}$ at the point $\mathrm{Q}\left(k_1,k_2\right)$, then $k_1+k_2$ is equal to :

1. A.2
2. B.$\frac{4}{7}$
3. C.$\frac{2}{7}$
4. D.4

Let $\mathrm{A}(\alpha ,-2)$, $\mathrm{B}(\alpha ,6)$ and $\mathrm{C}\left(\frac{\alpha }{4},-2\right)$ be vertices of a $△\mathrm{ABC}$. If $\left(5,\frac{\alpha }{4}\right)$ is the circumcentre of $△\mathrm{ABC}$, then which of the following is NOT correct about $△\mathrm{ABC}$?

1. A.area is 24
2. B.perimeter is 25
3. C.circumradius is 5
4. D.inradius is 2

The equations of the sides $\mathrm{AB}$, $\mathrm{BC}$ and $\mathrm{CA}$ of a triangle $\mathrm{ABC}$ are $2x+y=0$, $x+py=39$ and $x-y=3$ respectively and $\mathrm{P}(2,3)$ is its circumcentre. Then which of the following is NOT true?

1. A.$(\mathrm{AC}{)}^2=9p$
2. B.$(\mathrm{AC}{)}^2+p^2=136$
3. C.$32<\text{area }(\Delta \mathrm{ABC})<36$
4. D.$34<\text{area }(\Delta \mathrm{ABC})<38$

A point $P$ moves so that the sum of squares of its distances from the points $(1,2)$ and $(-2,1)$ is 14. Let $f(x,y)=0$ be the locus of P, which intersects the $x$-axis at the points $A$, B and the $y$-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to :

1. A.$\frac{9}{2}$
2. B.$\frac{3\sqrt{17}}{2}$
3. C.$\frac{3\sqrt{17}}{4}$
4. D.9

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 $(2,a)$, $-\frac{1}{2}<a<2$, then p is equal to ____ .

Let ${\alpha }_1,{\alpha }_2({\alpha }_1<{\alpha }_2)$ be the values of $\alpha$ for the points $(\alpha ,-3)$, $(2,0)$ and $(1,\alpha )$ to be collinear. Then the equation of the line, passing through $({\alpha }_1,{\alpha }_2)$ and making an angle of $\frac{\pi }{3}$ with the positive direction of the x-axis, is :

1. A.$x-\sqrt{3}y-3\sqrt{3}+1=0$
2. B.$\sqrt{3}x-y+\sqrt{3}+3=0$
3. C.$x-\sqrt{3}y+3\sqrt{3}+1=0$
4. D.$\sqrt{3}x-y+\sqrt{3}-3=0$

Let a triangle be bounded by the lines $L_1:2x+5y=10$; $L_2:-4x+3y=12$ and the line $L_3$, which passes through the point $P(2,3)$, intersects $L_2$ at A and $L_1$ at B. If the point P divides the line-segment AB, internally in the ratio $1:3$, then the area of the triangle is equal to :

1. A.$\frac{110}{13}$
2. B.$\frac{132}{13}$
3. C.$\frac{142}{13}$
4. D.$\frac{151}{13}$

In an isosceles triangle ABC, the vertex A is $(6,1)$ and the equation of the base BC is $2x+y=4$. Let the point B lie on the line $x+3y=7$. If $(\alpha ,\beta )$ is the centroid of $\Delta \mathrm{ABC}$, then $15(\alpha +\beta )$ is equal to :

1. A.39
2. B.41
3. C.51
4. D.63

Let R be the point $(3,7)$ and let P and Q be two points on the line $x+y=5$ such that PQR is an equilateral triangle. Then the area of $\Delta \mathrm{PQR}$ is :

1. A.$\frac{25}{4\sqrt{3}}$
2. B.$\frac{25\sqrt{3}}{2}$
3. C.$\frac{25}{\sqrt{3}}$
4. D.$\frac{25}{2\sqrt{3}}$

If the two lines $l_1:\frac{x-2}{3}=\frac{y+1}{-2},z=2$ and $l_2:\frac{x-1}{1}=\frac{2y+3}{\alpha }=\frac{z+5}{2}$ are perpendicular, then an angle between the lines $l_2$ and $l_3:\frac{1-x}{3}=\frac{2y-1}{-4}=\frac{z}{4}$ is :

1. A.${\cos }^{-1}\left(\frac{29}{4}\right)$
2. B.${\sec }^{-1}\left(\frac{29}{4}\right)$
3. C.${\cos }^{-1}\left(\frac{2}{29}\right)$
4. D.${\cos }^{-1}\left(\frac{2}{\sqrt{29}}\right)$