Straight Lines Questions - JEE

Given line is $5x - y = 1$. A line $L$ is perpendicular to it. The area of the triangle formed by line $L$, line $5x - y = 1$, and the origin is 5 square units. The distance of line $L$ from origin is:

A ray of light is incident on a mirror. The normal at the point of incidence makes an angle of 30° with the horizontal. If the point of incidence is $(\sqrt{3}, 0)$ and a point (0, 1) lies on the ray of light (on the line), then the equation of the reflected ray is:

Three lines have slopes $m_1 = 5$, $m_2 = 3$, $m_3 = -1$ (arranged in descending order). The angles $A$, $B$, $C$ between consecutive lines satisfy:$\tan A = \dfrac{m_1 - m_2}{1 + m_1 m_2} = \dfrac{2}{1+15} = \dfrac{1}{8}$$\tan B = \dfrac{m_2 - m_3}{1 + m_2 m_3} = \dfrac{3+1}{1-3} = -2$$\tan C = \dfrac{m_3 - m_1}{1 + m_3 m_1} = \dfrac{-1-5}{1-5} = \dfrac{3}{2}$If $\displaystyle\sum \tan^2 A = \dfrac{1}{64} + 4 + \dfrac{9}{4} = \dfrac{p+q}{93}$, find $\dfrac{p+q}{93}$.

If a variable line drawn through the intersection of the lines $\dfrac{x}{3} + \dfrac{y}{4} = 1$ and $\dfrac{x}{4} + \dfrac{y}{3} = 1$ meets the coordinate axes at A and B, (A ≠ B), then the locus of the mid-point of AB is

The given lines $L_1$ and $L_2$ are parallel and the distance between them ($BC$ or $AD$) is $(15-5)/5 = 2$ units. A parallelogram $AA_1BB_1$ is formed. The area of the parallelogram is least for $\theta = \pi/4$. If the slope of $AB$ is $m$, then $1 = \left|\dfrac{m + 3/4}{1 - \dfrac{3m}{4}}\right|$. Find the slope $m$ and hence the equation of line $L$. The minimum area of the parallelogram $AA_1BB_1$ is:

Number of possible ordered pair(s) $(x, y)$ of all positions of point P on AB so that area of the rectangle PDOC is 30 sq. units is:

A triangle with vertices (4, 0), (−1, −1), (3, 5) is

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

A vertex of an equilateral triangle is $(2, 3)$ and the equation of the opposite side is $x + y = 2$. Find the equations of the other two sides and the length of each side of the triangle.

Let ABCD be a parallelogram, the equations of whose diagonals are $AC: x + 2y - 3 = 0$ and $BD: 2x + y - 3 = 0$. If the length of the diagonal $AC = 4$ units and the area of the parallelogram $[ABCD] = 8$ square units. The length of side BD is:

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

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:

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

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

The equation of the pair of bisectors of the angles between the pair of lines represented by $ax^2 + 2hxy + ay^2 = 0$ if the product of slopes of four lines represented by a given equation is 1 and a pair of lines represents the bisectors of angles between the other two, then the product of the slopes of each pair is:

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

Two lines $y = -\sqrt{3}x$ and $y = \sqrt{3}x$ intersect at the origin. A point $P(2, 2)$ is given. If $B$ is a point on the line $y = \sqrt{3}x$ and $K$ is on line $y = \sqrt{3}x$ such that $\angle AOB = 60°$ (where $O$ is origin and $\triangle OAB$ is equilateral), then $\angle KBP$ equals:

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

Straight Lines Questions (433)