Straight Lines — JEE Main PYQs

A man starts walking from the point $\mathrm{P}(-3,4)$, touches the x -axis at R , and then turns to reach at the point $\mathrm{Q}(0,2)$. The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $50(PR{)}^2+(RQ{)}^2$ is equal to

Let $A(-1,1),B(3,4)$ and $C(2,0)$ be given three points. A line $y=mx,m>0$, intersects lines $AC$ and BC at point P and Q respectively. Let ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ be the area of $△\mathrm{ABC}$ and $△PQC$ respectively, such tha …

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

The number of integral values of $m$ so that the abscissa of point of intersection of lines $3x+4y=9$ and $y=mx+1$ is also an integer, is:

Let $A(1,0),B(6,2)$ and $C\left(\frac{3}{2},6\right)$ be the vertices of a triangle $ABC$. If $P$ is a point inside the triangle ABC such that the triangles $\mathrm{APC},\mathrm{APB}$ and BPC have equal areas, then the length of the line segment PQ, where Q is the point …

The locus of the mid - points of the perpendiculars drawn from points on the line, $x=2y$ to the line $x=y$ is:

Let two points be $\mathrm{A}(1,-1)$ and $\mathrm{B}(0,2)$. If a point $\mathrm{P}\left({\mathrm{x}}^{\prime },{\mathrm{y}}^{\prime }\right)$ be such that the area of $\Delta PAB=5$ sq. units and it lies on the line, $3x+y-4\lambda =0$, then a value of $\lambda$ is:

Let C be the centroid of the triangle with vertices $(3,-1),(1,3)$ AND $(2,4)$. Let P be the point of intersection of the lines $x+3y-1=0$ and $3x-y+1=0$. Then the line passing through the points C and P also passes through the point:

If a $△\mathrm{ABC}$ have vertices $\mathrm{A}(-1,7),\mathrm{B}(-7,1)$ and $\mathrm{C}(5,-5)$, then its orthocenter has coordinates:

Let L denote the line in the xy -plane with x and y intercepts as 3 and 1 respectively. Then the image of the point $(-1,-4)$ in this line is:

If the line, $2x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4x-2y+\alpha =0$ and $6x-3y+\beta =0$, respectively, then the sum of all possible values of $\alpha$ and $\beta$ is ______.

A ray of light coming from the point $(2,2\sqrt{3})$ is incident at an angle $30^{\circ }$ on the line $x=1$ at the point $A$. The ray gets reflected on the line $x=1$ and meets $x$-axis at the point $B$. Then, the line $AB$ passes through the point:

A triangle ABC lying in the first quadrant has two vertices as $\mathrm{A}(1,2)$ and $\mathrm{B}(3,1)$. If $\angle BAC=90^{\circ }$, and $\mathrm{ar}(△ABC)=5\sqrt{5}$ sq. units, then the abscissa of the vertex $C$ is:

If the perpendicular bisector of the line segment joining the points $\mathrm{P}(1,4)$ and $\mathrm{Q}(\mathrm{k},3)$ has $y$-intercept equal to -4 , then a value of $k$ is:

A point on the straight line, $3x+5y=15$ which is equidistant from the coordinate axes will lie only in:

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:

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

A point $P$ moves on the line $2x-3y+4=0$. If $Q(1,4)$ and $R(3,-2)$ are fixed points, then the locus of the centroid of $△\mathrm{PQR}$ is a line:

If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is:

Lines are drawn parallel to the line $4x-3y+2=0$ at a distance $\frac{3}{5}$ from the origin. Then which one of the following points lies on any of these lines?

If the straight line, $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta )$, then $\beta$ equals:

A triangle has a vertex at $(1,2)$ and the mid points of the two sides through it are $(-1,1)$ and $(2,3)$. Then the centroid of this triangle is :

A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of $60^{\circ }$ with the line $x+y=0$. Then an equation of the line $L$ is:

If a straight line passing through the point $\mathrm{P}(-3,4)$ is such that its intercepted portion between the coordinate axes is bisected at P , then its equation is:

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

Let S be the set of all triangle in the xy - plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is:

Let the equation 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:

Consider the set of all lines $px+qy+r=0$ such that $3p+2q+4r=0$. Which one of the following statements is true?

The region represented by $|x-y|\le 2$ and $|x+y|\le 2$ is bounded by a:

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

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 OPRQ is completed, then the locus of R is :

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

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?

Locus of the image of the point $(2,3)$ in the line $(2x-3y+4)+k(x-2y+3)=0,k\in R$, is a :

Let $a,b,c$ and d be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies in the fourth quadrant and is equidistant from the two axes then

Let PS be the median of the triangle with vertices $\mathrm{P}(2,2),\mathrm{Q}(6,-1)$ and $\mathrm{R}(7,3)$. The equation of the line passing through $(1,-1)$ and parallel to PS is

A ray of light along $x+\sqrt{3}y=\sqrt{3}$ gets reflected upon reaching $x$-axis, the equation of the reflected rays is

The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $(0,1)(1,1)$ and $(1,0)$ is

If the line $2x+y=k$ passes through the point which divides the line segment joining the points ( 1,1 ) and $(2,4)$ in the ratio $3:2$, then $k$ equals

The lines $L_1:y-x=0$ and $L_2:2x+y=0$ intersect the line $L_3:y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersect $L_3$ at $R$.
Statement -1 : The ratio $\mathrm{PR}:\mathrm{RQ}$ equals …

The line $L$ given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line $K$ is parallel to $L$ and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is

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

The perpendicular bisector of the line segment joining $\mathrm{P}(1,4)$ and $\mathrm{Q}(\mathrm{k},3)$ has y-intercept -4. Then a possible value of k is

Let $A(h,k)$, $B(1,1)$ and $C(2,1)$ be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is 1, then the set of values which 'k' can take is given by

Let $P=(-1,0)$, $Q=(0,0)$ and $R=(3,3\sqrt{3})$ be three points. The equation of the bisector of the angle $PQR$

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

A straight line through the point $\mathrm{A}(3,4)$ is such that its intercept between the axes is bisected at A. Its equation is

If (a, $a^2$) falls inside the angle made by the lines $y=\frac{x}{2},x>0$ and $y=3x,x>0$, then a belongs to

The line parallel to the $x$-axis and passing through the intersection of the lines ax + $2by+3b=0$ and $bx-2ay-3a=0$, where $(a,b)\ne (0,0)$ is

If a vertex of a triangle is $(1,1)$ and the mid-points of two sides through this vertex are $(-1,2)$ and $(3,2)$, then the centroid of the triangle is