Straight Lines Questions - JEE

Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AM N is 4 9 of the area of the triangle OAB and AN : NB = \lambda : 1, then the sum of all possible value(s) of is \lambda :

Two sides a and b of triangle ABC are given by the roots of the equation x2 - 4x + 1 = 0 and the included angle between them is $\frac{\pi}{3}$ then the value of $\left(\frac{c^{2}}{a+b}\right)$ is :

Let the points $\left(\dfrac{11}{2},\alpha\right)$ lie on or inside the triangle with sides $x+y = 11$, $x+2y = 16$ and $2x+3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to:

Let $\alpha,\beta,\gamma,\delta\in\mathbb{Z}$ and let $A(\alpha,\beta)$, $B(1,0)$, $C(\gamma,\delta)$ and $D(1,2)$ be the vertices of a parallelogram $ABCD$. If $AB=\sqrt{10}$ and the points $A$ and $C$ lie on the line $3y=2x+1$, then $2(\alpha+\beta+\gamma+\delta)$ is equal to

Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If the centroid of △PQR is the point (\alpha, \beta), then 15(\alpha - \beta) is equal to :

Let the lines 3x - 4y - \alpha = 0, 8x - 11y - 33 = 0, and 2x - 3y + \lambda = 0 be concurrent. If the image of the point in the line 2x - 3y + \lambda = 0 is ( , then |\alpha\lambda| is equal to 57 -40 (1, 2) , ) 13 13

Let $R$ be the interior region between the lines $3x-y+1=0$ and $x+2y-5=0$ containing the origin. The set of all values of $a$, for which the points $(a^2,a+1)$ lie in $R$, is:

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

A square of side $a$ lies above the $x$-axis and has one vertex at the origin. The side passing through the origin makes an angle \(\alpha\left(0

$I(1,0)$ is the centre of circle of triangle $ABC$, the equation of $BI$ is $x - 1 = 0$ and equation of $CI$ is $x - y - 1 = 0$, then angle $BAC$ is:

Given $A \equiv (1,1)$ 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:

Let $R$ be rectangle $x=0,x=2,y=0,y=5$. $A(\alpha,0)$, $B(0,\beta)$ divide rectangle area in ratio $4:1$. Midpoint of $AB$ lies on a

The vertices of a triangle are A(4, 0), B(−1, −1), C(3, 5). The triangle is

A line $y = m(x-6)$ is tangent to a curve such that the distance from the point $(5, 1)$ to the line $mx - y - 6m = 0$ is 5. If $\frac{p}{q} = \frac{8}{15}$ in lowest terms, find $p + q$.

Locus of the centroid of the variable triangle OAB has the equation (where 'O' is the origin):

If the line $2x + y = k$ passes through the point which divides the line segment joining the points $(1, 1)$ and $(2, 4)$ in the ratio $3 : 2$, then $k$ equals

The equation of a pair of straight lines is $ax^2 + 2hxy + by^2 = 0$. By what angle must the axes be rotated so that the term containing $xy$ in the equation may be removed?

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

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:

If the equation of the locus of a point equidistant from the points $(a_1, b_1)$ and $(a_2, b_2)$ is $(a_1 - a_2)x + (b_1 - b_2)y + c = 0$, then the value of $c$ is

Find the value of x_1, if the distance between the points (x_1, 2) and (3, 4) be 8.

A ray of light coming from the point (1, 2) is reflected at a point A on the x-axis and then passes through the point (5, 3). The equation of the line containing the incident ray is:Given: The image of the point $\left(\frac{1}{2}, 0\right)$ lies on the incident ray and the equation of the line of incidence of the ray of light is $41x - 38y + 38 = 0$.

In a triangle $ABC$, coordinates of $A$ are $(1, 2)$ and the equations of the medians through $B$ and $C$ are respectively $x + y = 5$ and $x = 4$. Then area of $\triangle ABC$ (in sq. units) is

Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

Find the value of m such that the points (1, 2), (2, -1), and (−1, 3) are collinear.

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:

Let $A(1,0)$, $B(2,-1)$ and $C\left(\dfrac{7}{3},\dfrac{4}{3}\right)$ be three points. If the equation of the bisector of the angle $ABC$ is $\alpha x+\beta y=5$, then the value of $\alpha^2+\beta^2$ is

Let $\triangle ABC$ be an isosceles triangle with $AB = AC$. If $AB: 4x + y = 7$, $AC: x + 4y = 7$ and $BC$ is passing through $(1, 1)$, then possible equation of $BC$ is:

If one of the lines given by the equation $2x^2 + pxy + 3y^2 = 0$ coincide with one of those given by $2x^2 + qxy - 3y^2 = 0$ and the other lines represented by them be perpendicular, then

If a point moves such that it is always at equal distance from lines $AB$, $BC$ and $AC$ are drawn which divides the triangle into six regions. If the area of three triangles formed in these regions are 9, 16 and 25 square units, then the area (in sq. units) of $\triangle ABC$ is

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 lines $x + y - 1 = 0$, $(m-1)x + (m^2 - 7)y - 5 = 0$ and $(m-2)x + (2m-5)y = 0$ are

The coordinates of the vertices are $O$, $P$, $Q$, $R$ as $(0, 0)$, $(a, 0)$, $(a, a)$, $(0, a)$ respectively. Find the ratio of the area of $\triangle OMN$ to the area of the square, where $M$ is at $(a, \frac{a}{2})$ and $N$ is at $(\frac{3a}{4}, a)$.

Find the area of the square ABCD where A(a,0), B(0,a), C(a,a), and the configuration involves point F at \left(-\frac{a}{3}, a\right) with BF = \frac{a}{4}(a + BF).

If one diagonal of a square is the portion of the line \frac{x}{a} + \frac{y}{b} = 1 intercepted by the axes, then the extremities of the other diagonal of the square are:

The coordinates of P are given by (P lies on the curve y = \log_{1/2}(x - 0.5) + \log_2\sqrt{4x^2 - 4x + 1} and on the circle x^2 + y^2 = 10)

Suppose ABC is a triangle with 3 acute angles A,B and C. The point whose coordinates are (\cos B - \sin A, \sin B - \cos A) can be in the -

The points (2, 1) and (-3, 5) lie along the line $3x - 2y + 1 = 0$ on which side?

Let A(1, 2), B(3, 4) be two points and C(x, y) be a point such that $(x-1)(x-3) + (y-2)(y-4) = 0$. If area of $\triangle ABC$ is 1 sq unit, then the maximum number of positions of C in the XY-plane, is

Find the value of k if the mid-point of A(2, 5) and B(5, 1) lies on the line y = 2x + k.

If a point R(4, y, z) lies on the line segment joining the points P{2, -3, 4) and Q(8, 0, 10), then the distance of R from the origin is

If P = (1/x_p, p); Q = (1/x_q, q); R = (1/x_r, r) where x_k ≠ 0, denotes the k-th terms of a H.P. for k ∈ ℕ, then:

Let PQR be a right angled isosceles triangle, right angled at P(2, 1). If the equation of the line QR is $2x + y = 3$, then the equation representing the pair of lines PQ and PR is:

Area of a triangle is 5 sq units and two of its vertices are (2, 1) and (3, -2). If its third vertex is on the line y = x + 3, then it is

What is the y-intercept of the line that is parallel to y = 3x, and which bisects the area of a rectangle with corners at (0, 0), (4, 0), (4, 2) and (0, 2)?

If the pair of lines 6x² - axy - 3y² - 24x + 3y + b = 0 intersect on x-axis, then find the value of 20a - b.

Let $ A(x_1, y_1) $, $ B(x_2, y_2) $ and $ C(x_3, y_3) $ be the vertices of a triangle such that algebraic sum of perpendicular distance from $ A $, $ B $ and $ C $ to the variable line $ ax + by + c = 0 $ is always '0'. If $ 3a + 2b + c = 0 $, then find the value of $ \displaystyle\sum_{i=1}^{3} (x_i + y_i) $.

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 $(3h + 2k)$.

$\begin{vmatrix} 1 & 1 & -1 \\ p & 2 & 1 \\ 4 & 2p & 7 \end{vmatrix} = 0$

Straight Lines Questions (433)