Straight Lines Questions - JEE

The straight lines $3x + y - 4 = 0$, $x + 3y - 4 = 0$ and $x + y = 0$ form a triangle which is:

The coordinate axes rotated through an angle 135°. If the coordinates of a point P in the new system are known to be (4, -3), then the coordinates of P in the original system are

If a triangle ABC has vertices A(-1, 7), B(-7, 1) and C(5, -5), then its orthocentre has coordinates

If $(ax_1 + by_1 + c) + (ax_2 + by_2 + c) + (ax_3 + by_3 + c) = 0$, show that the line $ax + by + c = 0$ passes through the centroid of triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$. Find the value $15$ times the number of such conditions.

Let the equation $x^3 + y^3 + 3xy = 1$ represents the coordinate of one vertex $A$ and the equation of side $BC$ of the triangle $ABC$. If $B$ is the orthocentre of the triangle $ABC$, then the equation of side $AB$ is $y = mx + c$. Then absolute value of $(4 - m - c)$, 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° with the line x + y = 0. Then, an equation of the line L is

Given A ≡ (1, 1) and AB is any line through it cutting the x-axis in B. If AC is perpendicular to AB and meets the y-axis in C, then the equation of locus of mid-point P of BC is:

The equation of line passing through (1, 2) with slope m is y − 2 = m(x − 1). The area of triangle OPQ (where O is origin) is least when m equals:

Find the coordinates of vertex $C$ if the centroid is at $G = (1, \frac{5}{3})$, with vertices $A(\frac{11}{4}, \frac{4}{3})$ and $B(-\frac{11}{3}, \frac{1}{3})$ given.

Given that the equation $\sqrt{3}x + y = 1$ has slope angle 120°, any line with inclination of 60° with the above line has slope angle 60°. The equation of such a line passing through the point (3, −2) is:

The point $(x-3, y-3)$ satisfies $\dfrac{x-3}{\cos(\pi/4)} = \dfrac{y-3}{\sin(\pi/4)} = -2\sqrt{2}$. Find the new position of the point.

Area of the triangle formed by the lines through point (6, 0) and at a perpendicular distance of 5 from point (1, 3) and line $y = 16$ in square units is:

$m, n$ are integer with $0 < n < m$. $A$ is the point $(m, n)$ on the Cartesian plane. $B$ is the reflection of $A$ in the line $y = x$. $C$ is the reflection of $B$ in the $y$-axis, $D$ is the reflection of $C$ in the $x$-axis and $E$ is the reflection of $D$ in the $y$-axis. The area of the pentagon $ABCDE$ is:

In a $\triangle ABC$, 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

A line passing through the point $A(9,0)$ makes an angle of $30°$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of $15°$ in the clockwise direction, then its equation in the new position is

$ABC$ is a variable triangle with the fixed vertex $C(1, 2)$ and $A$, $B$ having coordinates $(\cos t, \sin t)$, $(\sin t, -\cos t)$ respectively, where $t$ is a parameter. Find the locus of the centroid of $\triangle ABC$.

A line L has intercepts a and b on the coordinate axes. Keeping the origin fixed, the axes are rotated through a fixed angle. Now, the same line has intercepts p and q on the new axes. Then which relation holds?

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?

Straight Lines Questions (433)