Straight Lines Questions - JEE

In triangle ABC with H as orthocenter at origin (0, 0), B = (-2, 3), and C = (5, -1), find the coordinates of vertex A.

Question 86: Statement-1: If the system of equations $2x + 3y = a$ and $bx + 4y = 5$ has infinite solutions, then $a = \frac{15}{8}$, $b = \frac{8}{5}$.Statement-2: Straight lines $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$ are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.

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 k/h equals

Without changing the direction of coordinate axes, the origin is shifted to (h, k), then from the equation x^2 + y^2 - 4x + 6y - 7 = 0 the terms containing linear powers are missing. Then, point (h, k) is

The distance between the lines $5x - 12y + 65 = 0$ and $10x - 24y - 39 = 0$ is

ABC is an isosceles triangle. If the coordinates of the base are B(1, 3) and C(-2, 7), the coordinates of vertex A is

If the points $(0, 0)$, $(2, 2\sqrt{3})$ and $(a, b)$ are the vertices of an equilateral triangle, then $(a, b)$ is

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

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

If $L = \left(\frac{1}{x_l}\right), M = \left(\frac{1}{x_m}\right), N = \left(\frac{1}{x_n}\right)$ where $x_k \neq 0$, denotes the $k^{th}$ terms of a H.P. for $k \in N$, then:

The orthocentre of the triangle with vertices $(5, 0)$, $(0, 0)$, $\left(\frac{5}{2}, \frac{5\sqrt{3}}{2}\right)$ is:

The locus of centroid of the triangle whose vertices are (a cos t, a sin t) (b sin t, - b cos t) and (1, 0) where t is a parameter, is :

Find the value of the y-intercept $c$ of the line $y = 2x + c$ such that the perpendicular distance from the origin to this line equals 5.

The value of $a+b+c$ equals:

Let the equations 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:

Which of the following can't be the vertex of the triangle:

Which of the following can be possible orthocenter of the triangle:

For what values of λ the following three lines are concurrent?$x + y = 1$, $\lambda x + 2y = 3$, $\lambda^2 x + 4y + 9 = 0$

In a $\triangle ABC$, the vertex $A$ is $(1, 1)$ and orthocenter is $(2, 4)$. If the sides $AB$ and $BC$ are members of the family of straight lines $ax + by + c = 0$. Where $a, b, c$ are in A.P., then the coordinates of vertex $C$ are $(h, k)$. The value of $2h + 9k$ is ________.

Two equal sides $OA$ and $OB$ of an isosceles triangle lie in the first quadrant. If the slopes of $OA$ and $OB$ are $\frac{7}{17}$ and $1$, respectively and the length of perpendicular from $O$ to $AB$ is $\sqrt{13}$, if the equation of the side $AB$ is $ax + by = c$ then the value of $(c - a - b)$ is __________.

A line intersects the x-axis at $A(7, 0)$ and y-axis $B(0, -5)$. A variable line $PQ$ perpendicular to $AB$ intersects the x-axis at $P$ and the y-axis at $Q$. If $AQ$ and $BP$ intersect at $R$, show that the locus of $R$ is $x^2 + y^2 - ax + by = 0$. Then the value of $(a - b)$ is ______.

In a triangle $ABC$, the coordinates of $A$ is $(1, 2)$ and the equations to the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$. If coordinate of $B$ is $(x_1, y_1)$ and co-ordinate of $C$ is $(x_2, y_2)$ then $x_1y_2 + x_2y_1$ equals____.

If the equal sides $PQ$ and $PR$ (each equal to 2) of a right angled isosceles $\triangle PQR$ be produced to $A$ and $B$ so that $OA.RB = PR^2$ then the line $AB$ passes through a fixed point which also satisfies the line $ax + by - 6 = 0$ then $a + b$ is ____.

If from point $P$ (4, 4) perpendiculars to the straight lines $3x + 4y + 5 = 0$ and $y = mx + 7$ meet at $Q$ and $R$ respectively and area of triangle $PQR$ is maximum. Then the value of $12m$ must be ____.

The number of possible straight lines, passing through (2, 3) and forming a triangle with coordinate axes, whose area is 12 sq. units, is:

Let a line passing through point A divides the square ABCD into two parts so that area of one portion is double the other, then the length of portion of line inside the square is:

The line (k + 1)2x + ky - 2k^2 - 2 = 0 passes through a point regardless of the value k. Which of the following is the line with slope 2 passing through the point ?

If $6a^2 - 3b^2 - c^2 + 7ab - ac + 4bc = 0$, then the family of lines $ax + by + c = 0$ is concurrent at ordered pairs $(A, B)$ and $(C, D)$. $A > 0$ then the value of $A - B - C - D$ is ______.

If an equilateral triangle has one vertex at the point (0, 0) and another at (3, \sqrt{3}), then the coordinates of the third vertex is

Given A (0, 0) and B (x, y) with x ∈ (0, 1) and y > 0. Let the slope of the line AB equals m₁. Point C lies on the line x = 1 such that the slope of BC equals m₂ where 0

Two opposite vertices of a rectangle are (1, 3) and (5, 1). If the rest two vertices lie on the line y - x + l = 0, then l is equal to

The ordered pair (x, y) is, where H(x, y) are the coordinates of the orthocentre of triangle ABC with vertices A(9, 3), B(7, –1) and C(1, –1).

A straight line L through the point (3, −2) is inclined at an angle of 60° to the line $\sqrt{3}x + y = 1$. If L also intersects the x-axis, then the equation of L is

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?

Straight Lines Questions (433)