Straight Lines Questions - JEE

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 $(3h + 2k)$.

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

P (3, 1), Q (6, 5) and R (x, y) are three points such that angle PRQ is right angle and the area of △PRQ is 7, then number of such points R is.

Let the lengths of the altitudes from the vertices $A(-1,1)$, $B(5,2)$, $C(3,-1)$ of $\triangle ABC$ are $p_1$, $p_2$, $p_3$ units respectively, then the value of $\frac{(z_1)^{-1} + (z_2)^{-1}}{(z_3)^{-1}}$ is equal to

Let the opposite angular points of a square be (3, 4) and (1, -1). Then, the coordinates of the remaining angular points are

A ray of light passing through the point (2, 1) is reflected on the line $y = 1$. The equation of the reflected ray satisfies $SS_1 = T^2$, i.e., $(x^2 + y^2 - 1) \cdot 4 = (2x + y - 1)^2$, leading to $3y^2 - 4xy + 4x + 2y - 5 = 0$. Find the slope of the reflected ray.

The ratio of length of segments A1A2 and A1A3 is

The equations of two sides $AB$ and $AC$ of a triangle $ABC$ are $4x+y=14$ and $3x-2y=5$, respectively. The point $\left(2,-\dfrac{4}{3}\right)$ divides the third side $BC$ internally in the ratio $2:1$. The equation of the side $BC$ is

Let equation of line is $\frac{x}{a} + \frac{1}{a} = 1 = \frac{x}{a} + \frac{y}{b} - 1$. If both intercepts are positive, then find the sum of intercepts is equal to?

A line passes through the point of intersection of $\dfrac{x}{3} + \dfrac{y}{4} - 1 = 0$ and $\dfrac{x}{4} + \dfrac{y}{3} - 1 = 0$. The intercepts on the axes are A and B, and the mid-point of AB is (h, k). Then:

Each side of a square is of length 4 units. The center of the square is at (3, 7) and one of the diagonals is parallel to the line $y = x$. If the vertices of the square be (x₁, y₁), (x₂, y₂), (x₃, y₃) and (x₄, y₄) then find the value of $\max(y_1, y_2, y_3, y_4) - \min(x_1, x_2, x_3, x_4)$.

The equation of bisectors of lines xy = 0 are y = ±x. If these lines are contained in the pair of lines \[my^2 + (1 - m^2)xy - mx^2 = 0,\] then what is the value of m?

The area (in sq. units) enclosed by the graphs of $|x + y| = 2$ and $|x| = 1$ is

(A) Area of the parallelogram formed by the lines $y = mx$, $y = mx + 1$, $y = nx$ and $y = nx + 1$ equals:

Find the area of triangle with vertices at $A(1,13)$, $B(-4,1)$, $C(4,-5)$.

Given points $O(0,0)$ with $D$ as the mid-point of $BC$, find $D$ if $B = (1, \frac{8}{3})$.

Solving $y = x^2$ and $x^2 + (y - 2)^2 = 8$

$PQ = \sqrt{3^2 + 4^2} = 5$. Let length of altitude from $R$ to $h$. Then $\frac{1}{2} \times h \times 5 = 2$.

Number of values of $b$ for which in an acute triangle $ABC$, if the coordinates of orthocentre $'H'$ are $(4, b)$, centroid $'G'$ are $(b, 2b-8)$ and circumcentre $'S'$ are $(-4, 8)$ is

Straight Lines Questions (433)