Straight Lines Questions - JEE

It is desired to construct a right angled triangle $ABC$ ($\angle C = \pi/2$) in xy-plane so that its sides are parallel to co-ordinates axes and the medians through $A$ and $B$ lie on the lines $y = 3x+1$ and $y = mx+2$ respectively. The values of $m$ for which such a triangle is possible is/are:

Let $A(1,2)$ and $C(-3,-6)$ be two diagonally opposite vertices of a rhombus, whose sides $AD$ and $BC$ are parallel to the line $7x-y=14$. If $B(\alpha,\beta)$ and $D(\gamma,\delta)$ are the other two vertices, then $|\alpha+\beta+\gamma+\delta|$ is equal to

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 $(-4,-7)$. (S2): If positive numbers $2a,b,c$ are three consecutive terms of an A.P., then the lines $ax+by+c=0$ are concurrent at $(2,-2)$.

A piece of cheese is located at $(12, 10)$ in a coordinate plane. A mouse is at $(4, -2)$ and is running up the line $y = -5x+18$. At the point $(a, b)$, the mouse starts getting farther from the cheese rather than closer to it. The value of $(a+b)$ is:

Let $A=(1,2)$ and $B$ any point on $x^2+y^2=16$. If $P$ divides $AB$ in ratio $3:2$ and centre of locus of $P$ is $C(\alpha,\beta)$, then $|AC|$ is equal to

Two points $P_1$ and $P_2$ are at distances $r_1$ and $r_2$ respectively from the origin $O$ and $OP_1$ and $OP_2$ makes angle $\theta_1$ and $\theta_2$ respectively with the x-axis. Let there be a point $P$ on $P_1P_2$ such that $OP$ makes an angle $\frac{\theta_2 + \theta_1}{2}$ with the x-axis. Then $OP$ is:

A rectangle is formed by the lines $x=0,y=0,x=3$ and $y=4$. Let the line $L$ be perpendicular to $3x+y+6=0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $\left(\dfrac{1}{2},-5\right)$ from the line $L$ is equal to:

If $\frac{a}{\sqrt{bc}} - 2 = \sqrt[3]{\frac{b}{c}} + \sqrt[3]{\frac{c}{b}}$ where $a, b, c > 0$, then family of lines $\sqrt{ax} + \sqrt{by} + \sqrt{c} = 0$ passes through the point:

In a triangle $ABC$, if $A(2, -1)$ and $7x-10y+1=0$ and $3x-2y+5=0$ are equations of an altitude and an angle bisector respectively drawn from $B$, then equation of $BC$ is:

If $(\alpha,7\sqrt{3})$ lies on the curve traced by midpoints of $x\cos\theta+y\sin\theta=7$, $\theta\in(0,\frac{\pi}{2})$ between axes, then $\alpha$ is equal to

The area of the parallelogram formed by the lines 3x + 4y = 7a; 3x + 4y = 7b; 4x + 3y = 7c and 4x + 3y = 7d is-

The equations of two sides of a variable triangle are $x = 0$ and $y = 3$, and its third side is a tangent to the parabola $y^2 = 6x$. The locus of its circumcentre is:

Consider the triangle $OAB$ where $O = (0,0), B(3,4)$. If orthocenter of triangle is $H(1, 4)$, then coordinates of $A'$ is:

Number of lines that can be drawn through the point (4,–5) so that its distance from (–2,3) will be equal to 12 is equal to-

The distance between the two parallel lines is 1 unit. A point 'A' is chosen to lie between the lines at a distance 'd' from one of them. Triangle $ABC$ is equilateral with $B$ on one line and $C$ on the other parallel line. The length of the side of the equilateral triangle is:

The point (2, 1) is translated parallel to the line L : x - y = 4 by $2 \sqrt{3}$ units. If the new points Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is:

Area of square formed by the lines x2 y2 -2xy2 - 3y2 - 4x2 y + 8xy + 12y = 0 is :

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

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 $PS$ 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 $PS$ is

Given equation of straight line 2x − 3y + 5 = 0 is perpendicular to the line passing through point (7, 17). If this perpendicular line also passes through point (15, β), find the value of β.

The image of P (a, b) in the line x + y = 0 is Q and the image of Q in the line x - y = 0 is R, then the mid-point of PR is :

If a line intercepted between the coordinate axes is trisected at a point $A(4, 3)$, which is nearer to x-axis, then its equation is

Let B and C be the two points on the line y + x = 0 such that B and C are symmetric with respect to the origin. Suppose A is a point on y - 2x = 2 such that $\triangle$ABC is an equilateral triangle. Then, the area of the $\triangle$ABC is

The x-intercept of angle bisector of angle between the lines 2x + y - 2 = 0 and 2x + 4y + 7 = 0 which contains the fixed point on the family of lines (2cos$\alpha$ + 3 sin$\alpha$)x + (3 cos$\alpha$ - 5 sin$\alpha$)y = 5 cos$\alpha$ - 2 sin$\alpha$ for different values of $\alpha$, is equal to :

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

If the three distinct lines $x + 2ay + a = 0$, $x + 3by + b = 0$ and $x + 4ay + a = 0$ are concurrent, then the point $(a, b)$ lies on a

A line cuts the x-axis at A(α, 0) and the y-axis at B(0, β). A point P(4, 3) divides AB in the ratio 2:1. The equation of the line is:

The two adjacent sides of a parallelogram are given by 2x2 + 2y2 - 5xy = 0 and 5x + 2y = 1 is one of its diagonal. If area of parallelogram is S, then $\sqrt{S}$ is equal to :

The coordinates of a point dividing the line segment joining (1, 2) and (4, 5) externally in the ratio 2:1 is ________.

Line $L_1$ passes through $P(1, 2)$ and the portion of $L_1$ intercepted between the axes is bisected at $P(1, 2)$. Line $L$ is perpendicular to $L_1$ and passes through $(-2, 1)$. The point of intersection of $L_1$ and $L$ is:

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

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

Straight Lines Questions (433)