Straight Lines Questions (433)

Given a triangle whose vertices are at \(0, 0\), \(4, 4\) and \(10, 0\). A square is drawn in it such that its base is on the x-axis and its two corners are on the 2 sides of the triangle. The area of the square is equal to:

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)?

The base of an equilateral triangle is along the line given by \(3x + 4y = 9\). If a vertex of the triangle is \((1, 2)\), then the length of a side of the triangle is

Find the area of the pentagon \(ABCDE\) where \(A = (1, 3)\), \(B = (-2, 5)\), \(C = (-3, -1)\), \(D = (0, -2)\) and \(E = (2, 1)\).

A straight line through the origin $O$ meets the parallel lines $3x - 4y = 6$ and $6x - 8y + c = 0$ at points $Q$ and $P$ respectively such that $\left|\frac{OP}{OQ}\right| = \frac{4}{3}$. If $c = 2k$, then $k$ is equal to ______.

The point \((2a, a)\) lies on the line \(2x + 3y = 20\). The area of the triangle formed by the point \((a, a)\), origin, and point on the line is what?

Let $A(a,b)$, $B(3,4)$ and $(-6,-8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3,7b+5)$ from the line $2x+3y-4=0$ measured parallel to the line $x-2y-1=0$ is

In a rectangle with vertices at (0, 0), (a, 0), (a, b), and (0, b), if AD ⊥ BE where D and E are midpoints, find the relationship between a and b.

The portion of the line $4x+5y=20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of an angle between the lines $L_1$ and $L_2$ is:

The lines $x + y = 0$, $-4y = 0$ and $2x - y = 0$ are the altitudes of a triangle. If one of the vertices has coordinates of the form $\lambda, -\lambda)$, if the locus of the centroid of such a triangle is $ax + by = 0$, then the value of $a + b$ is ______.

If the straight line through the point $P(3, 4)$ makes an angle $\frac{\pi}{6}$ with the $x$-axis and meets the line $12x + 5y + 10 = 0$ at $Q$ then the value of $\frac{(12\sqrt{3} + 5)}{11}$PQ is ______.

Two roads are represented by the equation \(y - x = 6\) and \(x + y = 8\). An inspection bungalow has to be so constructed that it is at a distance of 100 from each of the roads. Possible location of the bungalow is given by

Straight lines $2x + y = 5$ and $x - 2y = 3$ intersect at point A. Points B and C are chosen on these two lines such that $AB = AC$. Then the equation of a line BC passing through the point $(2, 3)$ is:

The x-coordinates of the vertices of a square of unit area are the roots of the equation $x^2 - 3|x| + 2 = 0$ and the y-coordinates of the vertices are the roots of equation $y^2 - 3y + 2 = 0$, then the possible vertices of the square is/are:

The orthocentre of a triangle is at origin and circumcentre is at (2, –3). Then the centroid of the triangle is:

Let A = (0,0), B = (4,0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q.The locus of midpoints of all segments PQ as M varies along the segment AB is:

$ABC$ is a triangle, whose vertex $A$ is $(3,4)$, $L_1 = 0$, $L_2 = 0$ are the angle bisectors of angle $B$ and $C$ respectively where $L_1 = x + 2y - 5 = 0$, $L_2 = x - 2y - 3 = 0$ also $AB = KAI$ where $I$ is the incentre then $K$ is ____.

In a triangle $ABC$, the bisector of angles $B$ and $C$ lies along the lines $y = x$ and $y = 0$. If $A$ is $(1,2)$ then $\sqrt{10} d(A, BC)$ equal (where $d(A, BC)$ denotes the perpendicular distance of $A$ from $BC$).

Given a triangle whose vertices are at \((0, 0)\), \((4, 4)\) and \((10, 0)\). A square is drawn in it such that its base is on the x-axis and its two corners are on the 2 sides of the triangle. The area of the square is equal to:

Consider a trapezoid ABCD with AB = 8 cm perpendicular to the base, BC = 6 cm and AD = 10 cm. Distance of the point R lying on line AD from vertex A so that perimeter of triangle RBC is minimum is:

(B) Let PS be the median of the triangle with vertices P(2, 2), Q(6, -1) and R(7, 3). The equation of the line passing through (1, -1) and parallel to PS is:

A line through $A(-5, -4)$ meets the lines $x + 3y + 2 = 0$, $2x + y + 4 = 0$ and $x - y - 5 = 0$ at $B$, $C$ and $D$ respectively. If $\left(\frac{15}{AB}\right)^2 + \left(\frac{10}{AC}\right)^2 = \left(\frac{6}{AD}\right)^2$ equation of line is $2x + ly + c = 0$ then value of $2 + b + c$ is ______.

Let L_1: 3x + 4y = 1 and L_2: 5x - 12y + 2 = 0 be two given lines. Let the image of every point on L_1 with respect to a line L lies on L_2. Then a possible equation of L can be:

The vertices $B$ and $C$ of a triangle $ABC$ lie on the lines $3y = 4x$ and $y = 0$ respectively and the side $BC$ passes through the point $\left(\frac{2}{3}, \frac{1}{3}\right)$. If $ABOC$ is a rhombus, $O$ being the origin and the coordinates of $A$ are $(h, k)$, then $\frac{h}{k}$ is equal to ______.

Vertices of a parallelogram \(ABCD\) are \(A(3, 1)\), \(B(13, 6)\), \(C(13, 21)\) and \(D(3, 16)\). If a line passing through the origin divides the parallelogram into two congruent parts then the slope of the line is:

An altitude BD and a bisector BE are drawn in the triangle ABC from the vertex B. It is known that the length of side AC = 1, and the magnitudes of the angles \(\angle BEC\), \(\angle ABD\), \(\angle ABE\), \(\angle BAC\) form an arithmetic progression.Let B' be the image of point B with respect to side AC of \(\triangle ABC\), then the length BB' is equal to:

The area of triangle ABC is 20 cm². The coordinates of vertex A are (–5, 0) and B are (3, 0). The vertex C lies on the line x - y = 2. The coordinates of C are:

Consider a variable line 'L' which passes through the point of intersection 'P' of the lines \(3x + 4y - 12 = 0\) and \(x + 2y - 5 = 0\) meeting the coordinate axes at points A and B. Find the locus of the feet of the perpendicular from the origin on the variable line 'L':

Let $P$ be any point on the line $x - y + 3 = 0$ and $A$ be a fixed point $(3,4)$. If the family of lines given by the equation $(3\sec\theta + 5\cos e\theta)x + (7\sec\theta - 3\cos e\theta)y + 11(\sec\theta - \cos e\theta) = 0$ are concurrent at a point $B$ for all permissible values of $\theta$ and maximum value of $|PA - PB| = 2\sqrt{2n}$ ($n \in \mathbb{N}$), then find the value of $n$.

In a $\triangle ABC$, the equations of right bisectors of sides $AB$ and $AC$ are $3x + 4y = 20$ and $8x + 6y = 65$ respectively. If the vertex $A$ be $(10, 10)$, then the value of $\frac{1}{7}(ar\triangle ABC)$ is ________.

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:

Let m be a positive integer and let the lines \(13x + 11y = 700\) and \(y = mx - 1\) intersect in a point whose coordinates are integers. Then m equals to:

Find the number of integral values of m such that the line 3x + 4mx + 4 = 9 has specific intersection properties.

If the distance of any point $P(x, y)$ from the origin is defined as $d(x, y) = \max\{|x|, |y|\}$, $d(x, y) = 2$ then the area of curve represented by the locus of point $P$ is $S$ then $[S]$ is ____.

Example 40: The distance between two parallel lines is 1 unit. A point \(A\) is chosen to lie between the lines at a distance \(d\) from one of them. Triangle \(ABC\) is equilateral with \(B\) on one line and \(C\) on the other parallel line. The length of the side of the equilateral triangle is

If the vertices \(P\) and \(Q\) of a triangle \(PQR\) are given by \((2, 5)\) and \((4, -11)\) respectively, and the point \(R\) moves along the line \(N: 9x + 7y + 4 = 0\), then the locus of the centroid of the triangle \(PQR\) is a straight line parallel to:

A right angled triangle $ABC$ ($\angle C = \frac{\pi}{2}$) is constructed so that its sides are parallel to coordinate axes and the medians through $A$ and $B$ lie on the lines $y = 3x + 1$ and $y = mx + 2$ respectively. Then product of values of $m$ for which such a triangle is possible is ____.

Let $(3,4)$ be a fixed point. A straight line passing through this point cuts the positive direction of the coordinate axes at the points $P$ and $Q$. If $\lambda$ sq unit the minimum area of the triangle $OPQ$, $O$ being the origin. Then the value of $\lambda$ must be ____.

Let \(A = (0, 0)\), \(B = (5, 0)\), \(C = (5, 3)\) and \(D = (0, 3)\) are the vertices of rectangle ABCD. If P is a variable point lying inside the rectangle ABCD and \(d(P, L)\) denote perpendicular distance of point P from line L. If \(d(P, AB) \leq \min\{d(P, BC), d(P, CD), d(P, AD)\}\), then area of the region in which P lies is:

Given three points \(P\), \(Q\), \(R\) with \(P(5, 3)\) and \(R\) lies on the x-axis. If equation of \(RQ\) is \(x - 2y = 2\) and \(PQ\) is parallel to the x-axis, then the centroid of \(\triangle PQR\) lies on the line

A variable line '$L$' of the form $y = mx$ is drawn to meet the lines $L_1: 2x + 3y - 5 = 0$; $L_2: x + 2y - 5 = 0$ and $L_3: 6x + 4y - 5 = 0$ at points $A$, $B$ and $C$. A point $P(a,b)$ is taken on the line '$L$' also $\frac{k(a+b)}{OP} = \frac{1}{OA} + \frac{1}{OB} + \frac{1}{OC}$ then value of $k$ is ____.

If $x^2-y^2+2hxy+2gx+2fy+c=0$ is the locus of a point, which moves such that it is always equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$, then the value of $g+c+h-f$ equals

If D, E and F are the middle points of BC, CA and AB respectively then the area of the triangle DEF is:Consider 3 non-collinear points A(9, 3), B(7, –1) and C(1, –1).

A(6, 3), B(-3, 5), C(4, -2) and D(x, 3x) are four points. If the area of \triangle DBC and \triangle ABC are in the ratio 1:2, then x is equal to

If A(\cos a, \sin a), B(\sin a, -\cos a), C(1, 2) are the vertices of a \triangle ABC, then the locus of centroid of triangle is

(A) The incentre of the triangle with vertices (1, 3), (0, 0) and (2, 0) is:

251. Let \(f(x, y)\) be a locus of a point \(P(x, y)\) satisfying \(\alpha(2x - y + 1) + \beta(3x - y) + \gamma(2x + y - 5) = 0\) \(\forall\ \alpha, \beta, \gamma \in R\). The least distance between the curve \(f(x, y)\) and straight line \(3x - 4y + 19 = 0\) is:

Let the area of a $\triangle PQR$ with vertices $P(5,4)$, $Q(-2,4)$ and $R(a,b)$ be $35$ square units. If its orthocenter and centroid are $O\!\left(2,\dfrac{14}{5}\right)$ and $C(c,d)$ respectively, then $c+2d$ 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 $\triangle PQR$ is the point $(\alpha,\beta)$, then $15(\alpha-\beta)$ is equal to:

Suppose A(1, 1) and B(2, -3) are two points and D is a point on AB produced such that AD = 3AB. Then, coordinates of D are