Straight Lines Questions (433)

Let $\left(5,\dfrac{a}{4}\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$, $B(a,6)$ and $C\left(\dfrac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha+\beta+\gamma$ is

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

The combined equation of the two lines $ax + by + c = 0$ and $a'x + b'y + c' = 0$ can be written as $(ax + by + c)(a'x + b'y + c') = 0$. The equation of the angle bisectors of the lines represented by the equation $2x^2 + xy - 3y^2 = 0$ is

If A \(\equiv\) (3,1 + sin\(\theta\)) and B \(\equiv\) (-1, 1 + cos\(\theta\)), \(\theta\) \(\in\) R are two points and P is a point on the line x - y = 0 then minimum value of (PA + PB) is :

If the area of the triangle with vertices (x, 0), (1, 1) and (0, 2) is 4 sq units, then the value of x is

Let n Cr-1 = 28, n Cr = 56 and n Cr+1 = 70 . Let A(4 cos t, 4 sin t), B(2 sin t, -2 cos t) and C (3r - n, r - n - 1) 2 be the vertices of a triangle ABC , where t is a parameter. If (3x - 1) + (3y) = \alpha, is the locus of the centroid of 2 2 triangle ABC , then \alpha equals

Find the value of \(l\) such that point \(D(2, 1)\) lies on line \(AD\) where \(A(4, l)\) is given and the slope of \(AD\) is 2.

A man starts from the point \(P(-3, 4)\) and reaches point \(Q(0, 1)\) touching \(x\)-axis at \(R(\alpha, 0)\) such that \(PR + RQ\) is minimum, then \(5|\alpha|\) =

The equation of chord AB is y(l + 3) = 4(x + l). Show that this represents a family of lines passing through a fixed point, and find that point.

A light ray emits from the origin making an angle $30°$ with the positive x-axis. After getting reflected by the line $x + y = 1$, if this ray intersects x-axis at Q, then the abscissa of Q is

If \(x^{2}-y^{2}+2 h x y+2 g x+2 f y+c=0\) is the locus of a point, which moves such that it is always equidistant from the lines \(x+2 y+7=\) 0 and \(2 x-y+8=0\), then the value of \(g+c+h-f\) equals

In the adjacent figure $ABC$ is right angled at $B$. If $AB = 4$ and $BC = 3$ and side $AC$ slides along the coordinate axes in such a way that $B$ always remains in the first quadrant, then $B$ always lie on straight line:

Let $A(0,1)$, $B(1,1)$ and $C(1,0)$ be the midpoints of sides of a triangle with incentre $D$. If the focus of $y^2=4ax$ through $D$ is $(\alpha+\beta\sqrt{2},0)$, then $\dfrac{\alpha}{\beta^2}$ is equal to

Let ${}^nC_{r-1} = 28$, ${}^nC_r = 56$ and ${}^nC_{r+1} = 70$. Let $A(4\cos t, 4\sin t)$, $B(2\sin t,-2\cos t)$ and $C(3r-n, r^2-n-1)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $(3x-1)^2+(3y)^2 = \alpha$ is the locus of the centroid of triangle $ABC$, then $\alpha$ equals

Adjacent sides of parallelogram $ABCD$: $2x-3y=-23$, $5x+4y=23$. Diagonal $AC$: $3x+7y=23$. If $d$ is distance of $A$ from other diagonal $BD$, then $50d^2$ is equal to ______________

The bisectors of angle between the straight lines $y - b = \frac{2m}{1-m^2}(x-a)$ and $y - b = \frac{2m'}{1-m'^2}(x-a)$ are:

Let \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) be the vertices of triangle ABC. If angle C is obtuse, then the quantity \((x_3 - x_1)(x_3 - x_2) + (y_3 - y_1)(y_3 - y_2)\) is negative.

Let the diagonals of a convex quadrilateral ABCD intersect at point P, and let a, b, c, d denote the lengths of sides AB, BC, CD, and DA respectively. Then: Diagonals of quadrilateral ABCD are perpendicular if and only if \(a^2 + c^2 = b^2 + d^2\).

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:

A possible equation of L is:

If the orthocentre of the triangle formed by the lines $2x+3y-1=0$, $x+2y-1=0$ and $ax+by-1=0$ is the centroid of another triangle, whose circumcentre and orthocentre respectively are $(3,4)$ and $(-6,-8)$, then the value of $|a-b|$ is _____

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 ____.

Find the changed equation of the locus \(x^2 + 4xy + y^2 = 1\) when the lines \(x + y = 0\) and \(x - y + 1 = 0\) are taken as the new \(x\) and \(y\) axes respectively.

If mb > 0 and the line y = mx + b is given, determine the possible signs of m and b, and analyze the x-intercept.

Find the value of \(x\) when the slope between two points is \(\frac{0+2}{x-8} = \frac{4+2}{-3-8} = -\frac{6}{11}\).

Let \(x_1 = x_1,\ x_2 = x_1 r,\ x_3 = x_1 r^2\) and \(y_1 = y_1,\ y_2 = y_1 r,\ y_3 = y_1 r^2\). Which of the following is true about the points \(A \equiv (x_1, y_1)\), \(B \equiv (x_2, y_2)\), \(C \equiv (x_3, y_3)\)?

Let $A(-1,1)$ and $B(2,3)$ be two points and $P$ be a variable point above the line $AB$ such that the area of $\triangle PAB$ is 10. If the locus of $P$ is $ax+by=15$, then $5a+2b$ is:

In an acute triangle ABC, point H is the intersection point of altitude CE to AB and altitude BD to AC. A circle with DE as its diameter intersects AB and AC at points F and G respectively. If BC = 25, BD = 20 and BE = 7. The sum of the length of all the sides of △ABC is:

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 middle point of the segment AB:

Let the lines $3x-4y-\alpha = 0$, $8x-11y-33 = 0$, and $2x-3y+\lambda = 0$ be concurrent. If the image of the point $(1,2)$ in the line $2x-3y+\lambda = 0$ is $\left(\dfrac{57}{13},-\dfrac{40}{13}\right)$, then $|\alpha\lambda|$ is equal 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 $\Delta$ABC in the line 3x + 6y - 53 = 0. Then h 2 + k 2 + hk is equal to:

If $l_1:\ 3y-2x=3$ is the angular bisector of $l_2:\ x-y+1=0$ and $l_3:\ \alpha x+\beta y+17=0$, then $\alpha^2+\beta^2-\alpha-\beta$ is equal to ............

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 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