Straight Lines Questions - JEE

The circumcentre of a triangle lies at the origin and its centroid is the mid-point of the line segment joining the points $(a^2 + 1, a^2 + 1)$ and $(2a, -2a)$, $a \neq 0$. Then for any $a$, the orthocentre of this triangle lies on the line

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

Let $P(-1, 0)$, $Q(0, 0)$, $R(3, 3\sqrt{3})$ be three points then the equation of the bisector of the angle $\angle PQR$ is:

Which makes an angle of 135° with the axis of x and which cuts the axis of y at a distance -8 from the origin.

The centroid of a triangle is at G(h, k) and two vertices are C(0, 0) and O(h, k) with the centroid dividing a median in ratio 2:1. The locus of (h, k) is:

Find the equation of the line passing through the point P(1, 2) cutting the lines x + y − 5 = 0 and 2x − y = 7 at A and B respectively such that the harmonic mean of PA and PB is 10.

The distance of the point (1, 2) from the line $x + y + 5 = 0$ measured along the line parallel to $3x - y = 7$ is equal to:

The equation of line bisecting the obtuse angle between y − x = 2 and 2y + x = 5 is$$\frac{y - x - 2}{2} = \frac{x + 2y - 5}{n}$$where n is

If $A(3, 0)$ and $B(6, 0)$ are two fixed points and $U(a,b)$ is a variable point in the plane. $AU$ and $BU$ meet the y-axis at $C$ and $D$ respectively and $AD$ meet $OU$ at $V$. Then the coordinate of the point through which $CV$ always passes is:

If $ax + by = 1$ will be one of the bisectors of the given lines whose equations of bisectors are \[\frac{3x + 4y - 5}{5} = \pm\frac{5x - 12y - 10}{13}\] i.e., $64x - 8y = 115$, find the value of $52a + 5b$ (approximately).

Let A ≡ (3, 2) and B ≡ (5, 1). ABP is an equilateral triangle constructed on the side of AB remote from the origin then the orthocentre of triangle ABP is:

If $n_1$ is the number of points on the line $3x + 4y = 5$ which is at distance of $1 + \sin^2\theta$ units from (2, 3) and $n_2$ denotes the number of points on the line $3x + 4y = 5$ which is at distance of $\sec^2\theta + 2\csc^2\theta$ units from (1, 3), then find the sum of roots of equations $n_2x^2 - 6x + n_1 = 0$.

Let two straight lines drawn from the origin $O$ intersect the line $3x+4y=12$ at the points $P$ and $Q$ such that $\triangle OPQ$ is an isosceles triangle and $\angle POQ=90^\circ$. If $l=OP^2+PQ^2+QO^2$, then the greatest integer less than or equal to $l$ is:

The base of an isosceles triangle is the intercept made by the line $x + 2y = 4$ with the coordinate axes. If the equations of the equal sides be $x = 4$ and $y = mx + c$ then find the value of $8m + c$.

The given points are A = (0, 8/3), B = (1, 3), C = (82, 30). Then which of the following is true?

The ends of the base of an isosceles triangle are at $(2, 0)$ and $(0, 1)$ and the equation of one side is $x = 2$ then the orthocenter of the triangle is:

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

In a $\triangle A B C$, suppose $y=x$ is the equation of the bisector of the angle B and the equation of the side $A C$ is $2 x-y=2$. If $2 A B=B C$ and the point $A$ and $B$ are respectively $(4,6)$ and $(\alpha, \beta)$, then $\alpha+2 \beta$ is equal to

There exists two ordered triplets (a₁, b₁, c₁) and (a₂, b₂, c₂) for (a, b, c) for which the equation 4x² - 4xy + ay² + bx + cy + 1 = 0 represents a pair of identical straight lines in x-y plane. Find the value of a₁ + b₁ + c₁ + a₂ + b₂ + c₂.

If a line $L$ is perpendicular to the line $5x - y = 1$, and the area of the triangle formed by the line $L$ and the coordinate axes is 5, then the distance of line $L$ from the line $x + 5y = 0$ is

Let $A(1, 2)$ and $B(2, 3)$ be two points. A point $M$ lies on the line $x - y + 1 = 0$. Find the maximum value of $|MA - MB|$.

The equation of line $ AC $ in perpendicular form, given that $ A = (a\cos\alpha, a\sin\alpha) $ and the perpendicular from the origin makes an angle $ \left(\dfrac{\pi}{4} + \alpha\right) $ with the x-axis and has length $ \dfrac{a}{\sqrt{2}} $, is:

The foot of the perpendicular drawn from the origin, on the line, $3x + y = \lambda\,(\lambda \ne 0)$ is $P$. If the line meets $x$-axis at $A$ and $y$-axis at $B$, then the ratio $BP : PA$ is

If the pair of straight lines $x^2 - 2pxy - y^2 = 0$ and $x^2 - 2qxy - y^2 = 0$ be such that each pair bisects the angle between the other pair, then

If the point $M(h, k)$ lie on the line $2x + 3y = 5$ such that $|MA - MB|$ is maximum where $A(2, 3)$ and $B(1, 2)$, then find the value of $(3h + 2k)$.

Let $O = (0,0)$, $A = (2,0)$, $B = (1, \sqrt{3})$, and $M = (1,0)$. For a point $P$ in the plane, $d(P, OA) \leq \min[d(P,OB), d(P,AB)]$. The required area (of $\triangle OIA$ where $I$ is the incenter/centroid of the equilateral triangle $OAB$) equals $\dfrac{1}{2} \times OA \times IM$. Find this area (in sq. units).

The points (2, 5) and (5, 1) are two opposite vertices of a rectangle. If other two vertices are points on the straight line $y = 2x + k$, then the value of $k$ is:

The ratio in which the line segment joining (2, -3) and (5, 6) is divided by the x-axis is:

Let the intercepts be a and b such that a + b = −1. The line passes through the point (4, 3). How many equations of straight lines satisfy these conditions?

The area of the pentagon whose vertices are A(1, 1), B(7, 21), C(12, 2), D(7, -3) and E(0, -3) is

The distance of the point (1, 2) from the line $x + y + 5 = 0$ measured along the line parallel to $3x - y = 7$ is equal to:

The length of altitude through A of \triangle ABC, where A \equiv (-3, 0), B \equiv (4, -1), C \equiv (5, 2), is

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?

The base of an isosceles triangle is the intercept made by the line x + 2y = 4 with the coordinate axes. If the equations of the equal sides be x = 4 and y = mx + c, then find the value of 8m + c.

If the coordinates of the middle points of the sides of a triangle are (1, 1), (2, -3) and (3, 4), find the vertices and the centroid of the triangle.

The possible number of triangle is:

A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30° with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is

A ray of light coming from the point (2, 2√3) is incident at an angle 30° 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

If C is the reflection of A(2, 4) in X-axis and B is the reflection of C in Y-axis, then |AB| is equal to

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

Let A (-3, 2) and B (-2, 1) be the vertices of a triangle ABC. If the centroid of triangle ABC lies on the line 3x + 4y + 2 = 0, then the locus of vertex C is :

(A) (5,4)

Let y = m1x and y = m2x be two straight lines represented by x2 − 2cxy − 7y2 = 0. If (m1 + m2) = 4m1m2, then the value of c is:

Let ${ }^{n} {C}_{r-1}=28,{ }^{n} {C}_{r}=56$ and ${ }^{n} {C}_{r+1}=70$. Let $A(4 cos\ t, 4 sin \ t), {B}(2 sin \ t,-2$ cos t) and ${C}\left(3 r-n, r^{2}-n-1\right)$ be the vertices of a triangle ABC, where t is a parameter. If $(3 x-1)^{2}+(3 y)^{2}=\alpha$, is the locus of the centroid of triangle ABC, then $\alpha$ equals

If p and p′ be perpendiculars from the origin upon the straight lines $x \sec \theta + y \csc \theta = a$ and $x \cos \theta - y \sin \theta = a \cos 2\theta$ respectively, then the value of the expression $4p^2 + p'^2$ is

Let $ABCD$ be a parallelogram whose equations for the diagonals $AC$ and $BD$ are $x + 2y = 3$ and $2x + y = 3$, respectively. If length of diagonal $AC = 4$ units and area of parallelogram $ABCD = 8$ sq. units, then the length of other diagonal $BD$ is

Transform the equation $14x^2 - 4xy + 11y^2 - 36x + 48y + 41 = 0$ to the form $ax'^2 + by'^2 = 1$ by suitable change of axes.

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

If $D$, $E$ and $F$ are the middle points of $BC$, $CA$ and $AB$ respectively then the area of the triangle $DEF$ is:

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

Straight Lines Questions (433)