Straight Lines Questions (433)

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

If d(P, AB) ≥ max{d(P, BC), d(P, CD), d(P, AD)}, then area of the region in which P lies is:

If d(P, AB)² - 3/2 + d(P, AD)² ≥ 1, then area of region in which P lies is:

(B) 20

Two equal sides of an isosceles triangle are along -x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is :

The equation of the given line is 3x + 4y − 24 = 0. The incentre of the triangle formed by this line with the coordinate axes is:

Let $A(-2,-1)$, $B(1,0)$, $C(\alpha,\beta)$ and $D(\gamma,\delta)$ be the vertices of a parallelogram $ABCD$. If the point $C$ lies on $2x-y=5$ and the point $D$ lies on $3x-2y=6$, then the value of $|\alpha+\beta+\gamma+\delta|$ is equal to

If $(\alpha,\beta)$ is the orthocentre of $\triangle ABC$ with $A(3,-7)$, $B(-1,2)$, $C(4,5)$, then $9\alpha-6\beta+60$ is equal to

Let A(6, 8), B(10 cos \alpha, -10 sin \alpha) and C(-10 sin \alpha, 10 cos \alpha), be the vertices of a triangle. If L(a, 9) and G(h, k) be its orthocenter and centroid respectively, then (5a - 3h + 6k + 100 sin 2\alpha) is equal to ______ -.

A ray of light passing through the point \(A(1, 2)\) is reflected at a point \(B\) on the x-axis and then passes through \((5, 3)\). Then the equation of AB is:

Line $\frac{x}{a} + \frac{y}{b} = 1$ cuts the coordinate axes at $A(a, 0)$ and $B(0, b)$ and the line $\frac{x}{a'} + \frac{y}{b'} = -1$ at $A'(-a', 0)$ and $B'(0, -b')$. If the points $A$, $B$, $A'$, $B'$ are concyclic then the orthocentre of the triangle $ABA'$ is:

Let B' be the image of point B with respect to side AC of \(\triangle ABC\). Under the same conditions, the length BB' is equal to:

The true set of real values of \(\lambda\) for which the point P with co-ordinate \(\left(\lambda, \lambda^{2}\right)\) does not lie inside the triangle formed by the lines, x - y = 0; x + y - 2 = 0 and x + 3 = 0 is :

A triangle is formed by X-axis, Y-axis and the line $3x + 4y = 60$. Then the number of points $P(a, b)$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is ______.

A triangle is formed by the tangents at the point $(2, 2)$ on the curves $y^2 = 2x$ and $x^2 + y^2 = 4x$, and the line $x + y + 2 = 0$. If $r$ is the radius of its circumcircle, then $r^2$ is equal to ______.

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:

Let $ABC$ be a triangle formed by the lines $7x-6y+3 = 0$, $x+2y-31 = 0$ and $9x-2y-19 = 0$. Let the point $(h,k)$ be the image of the centroid of $\triangle ABC$ in the line $3x+6y-53 = 0$. Then $h^2+k^2+hk$ is equal to:

Two equal sides of an isosceles triangle are along $-x+2y = 4$ and $x+y = 4$. If $m$ is the slope of its third side, then the sum of all possible distinct values of $m$ is: