Straight Lines Questions (433)

Let $A$ be the point of intersection of the lines $3x+2y=14$, $5x-y=6$ and $B$ be the point of intersection of the lines $4x+3y=8$, $6x+y=5$. The distance of the point $P(5,-2)$ from the line $AB$ is

The number of integral values of b for which the origin and the point (1, 1) lie on the same side of straight line \(a^2x + aby + 1 = 0\) for \(a \in \mathbb{R} - \{0\}\) is.

Let ABC be a right triangle with \(\angle\)BAC = 90o then \(\left(\frac{r^{2}}{2 R^{2}}+\frac{r}{R}\right)\) is equal to : (where r and R have usual meaning in triangle.)

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(y₁, y₂, y₃, y₄) - min(x₁, x₂, x₃, x₄).

The equations of the sides AB, BC and CA of a triangle ABC are: $2x + y = 0$, $x + py = 21a$, $(a \neq 0)$ and $x - y = 3$ respectively. Let $P(2, a)$ be the centroid of $\triangle ABC$. Then $(BC)^2$ is equal to ______.

A straight line cuts off the intercepts $OA = a$ and $OB = b$ on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this line makes an angle of $\dfrac{\pi}{6}$ with positive direction of y-axis and the area of $\triangle OAB$ is $\dfrac{98}{3}\sqrt{3}$, then $a^2 - b^2$ is equal to:

The area of quadrilateral $ABCD$ with $A(2,1,1)$, $B(1,2,5)$, $C(-2,-3,5)$, $D(1,-6,-7)$ is equal to

Lines $l_1$ and $l_2$ through origin trisect the segment of $L:\ 9x+5y=45$ between axes. If $m_1,m_2$ are their slopes, the intersection of $y=(m_1+m_2)x$ with $L$ lies on

Let $(\alpha,\beta)$ be the centroid of $\triangle$ formed by $15x-y=82$, $6x-5y=-4$, $9x+4y=17$. Then $\alpha+2\beta$ and $2\alpha-\beta$ are roots of

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 $ABC$ be an isosceles triangle in which $A$ is at $(-1,0)$, $\angle A=\dfrac{2\pi}{3}$, $AB=AC$ and $B$ is on the positive $x$-axis. If $BC=4\sqrt{3}$ and the line $BC$ intersects the line $y=x+3$ at $(\alpha,\beta)$, then $\dfrac{\beta^4}{\alpha^2}$ is:

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $P(m,n)$ from the point $Q(-2,-3)$ is

If the sum of squares of all real values of $\alpha$, for which the lines $2x-y+3=0$, $6x+3y+1=0$ and $\alpha x+2y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is

The distance of the point $(2,3)$ from the line $2x-3y+28=0$, measured parallel to the line $\sqrt{3}x-y+1=0$, is equal to

If $P(6,1)$ be the orthocentre of the triangle whose vertices are $A(5,-2)$, $B(8,3)$ and $C(h,k)$, then the point $C$ lies on the circle:

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$, is $ax^2+by^2+cxy+dx+ey+170=0$, then the value of $a^2+2b+3c+4d+e$ is equal to:

Let the area of a △P QR with vertices P (5, 4), Q(-2, 4) and R(a, b) be 35 square units. If its orthocenter and centroid are O (2, 14 5 ) and C(c, d) respectively, then c + 2d is equal to

A rod of length eight units moves such that its ends A and B always lie on the lines x - y + 2 = 0 and y + 2 = 0 , respectively. If the locus of the point P , that divides the rod AB internally in the ratio 2 : 1 is 9 (x 2 + \alphay 2 + \betaxy + \gammax + 28y) - 76 = 0 , then \alpha - \beta - \gamma is equal to :

If the line segment joining the points $(5,2)$ and $(2,a)$ subtends an angle $\dfrac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is:

A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2 = 0$ and $y+2 = 0$, respectively. If the locus of the point $P$, that divides the rod $AB$ internally in the ratio $2:1$ is $9(x^2+\alpha y^2+\beta xy+\gamma x+28y)-76 = 0$, then $\alpha-\beta-\gamma$ is equal to:

Let a ray of light passing through the point $(3,10)$ reflects on the line $2x+y=6$ and the reflected ray passes through the point $(7,2)$. If the equation of the incident ray is $ax+by+1=0$, then $a^2+b^2+3ab$ is equal to _____

The vertices of a triangle are $A(-1,3)$, $B(-2,2)$ and $C(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:

If $A(1,-1,2)$, $B(5,7,-6)$, $C(3,4,-10)$ and $D(-1,-4,-2)$ are the vertices of a quadrilateral $ABCD$, then its area is:

Consider a triangle $ABC$ having the vertices $A(1,2)$, $B(\alpha,\beta)$ and $C(\gamma,\delta)$ and angles $\angle ABC=\dfrac{\pi}{6}$ and $\angle BAC=\dfrac{2\pi}{3}$. If the points $B$ and $C$ lie on the line $y=x+4$, then $\alpha^2+\gamma^2$ is equal to _____

Two vertices of a triangle $ABC$ are $A(3,-1)$ and $B(-2,3)$, and its orthocentre is $P(1,1)$. If the coordinates of the point $C$ are $(\alpha,\beta)$ and the centre of the circle circumscribing the triangle $PAB$ is $(h,k)$, then the value of $(\alpha+\beta)+2(h+k)$ equals

Given line x + y = 7 and point P(2, 3). Let point on line x + y = 7 where we draw perpendicular to point P(2, 3) be B(x1, y1) and point on line x + y = 7 from where point P(2, 3) is at distance 4 units be A(x2, y2). Find the slope of the line PA.

If \(\alpha, \beta, \gamma>0\) then the minimum value of the function f(x) = \(\sqrt{\alpha^{2}+x^{2}}+\sqrt{(x-\beta)^{2}+\gamma^{2}}\) is :

948. 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 \(\displaystyle\sum_{i=1}^{3}(x_i + y_i)\).

Two mutually perpendicular straight lines through origin form an isosceles triangle with the line 2x + y = 5, then the area of triangle is :

A rod of fixed length 2 slides along the coordinate axes. If it meets the axes at A(a, 0) and B(0, b), then the minimum value of \(\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2}\) equals : 

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 $AMN$ is $\dfrac{4}{9}$ of the area of the triangle $OAB$ and $AN:NB = \lambda:1$, then the sum of all possible values of $\lambda$ is:

If algebraic sum of distances of a variable line from points A(3, 0), B(0, 3) and C(- 3, - 3) is zero, then the line passes through the fixed point :

In a triangle $ABC$ if $AC = 3$, $BC = 4$ and median $AD$ and $BE$ are perpendicular to each other, $\triangle$ be the area of the triangle $ABC$. Then the value of $[\triangle]$ is ______ (where $[.]$ denotes the greatest integer).

A light beam emanating from the point A (3,10) reflects from the line 2x + y - 6 = 0 and then passes through the point B (5, 6). The equation of the incident and reflected beams are respectively :

A straight line cuts off the intercepts OA = a and OB = b on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this line makes an angle of \(\frac {\pi}6\) with positive direction of y-axis and the area of \(\triangle\)OAB is \(\frac{98}{3} \sqrt{3}\), then a2 - b2 is equal to:

Let S be the set of integral values of ‘a’ for which one root of quadratic equation (a2 - 2a + 3)x2 - 6ax + 4 = 0 is less than 1 while the other is greater than it, then the number of point(s) P (a2, a) where a \(\in\) S lies in the region between the lines x + y = 6 and x + y - 12 is (are) :

In the figure, \(P\) is a point inside a parallelogram \(ABCD\). The angle between the diagonals is \(\theta\) where \(\tan\theta = \left|\dfrac{-\frac{1}{2}+2}{1+1}\right| = \dfrac{3}{4}\), so \(\sin\theta = \dfrac{3}{5}\). The area of \(\triangle CPB\) is \(\dfrac{1}{2} \times PC \times PB \sin\theta = 2\), giving \(PB = \dfrac{10}{3}\). Find \(BD\).

If a parallelogram $ABCD$ is as shown in the figure. Let the distance between $AB$ & $CD$ is $d_1$, and the distance between $AD$ & $BC$ is $d_2$, and $\angle DAD = \angle DCB = \theta$ then

Let the angles made with the positive $x$-axis by two straight lines drawn from the point $P(2,3)$ and meeting the line $x+y=6$ at a distance $\sqrt{\dfrac{2}{3}}$ from the point $P$ be $\theta_1$ and $\theta_2$. Then the value of $(\theta_1+\theta_2)$ is:

Let B and C be the two points on the line $y + x = 0$ such that B and C are symmetric with respect to the origin. Suppose A is a point on $y - 2x = 2$ such that $\triangle ABC$ is an equilateral triangle. Then, the area of the $\triangle ABC$ is

If the orthocentre of the triangle, whose vertices are $(1, 2)$, $(2, 3)$ and $(3, 1)$ is $(\alpha, \beta)$, then the quadratic equation whose roots are $\alpha + 4\beta$ and $4\alpha + \beta$, is

The ordered pair $(\bar{x},\bar{y})$ is:

A chord $AB$ of length $\lambda$ moves so that $A,B$ are on a circle of radius $\lambda$. Locus of point dividing $AB$ in ratio $2:3$ is a circle of radius

In triangle ABC, if cos \(\frac{A}{2}=\frac{1}{2} \sqrt{\frac{b}{c}+\frac{c}{b}}\), then : [Note : All symbols used have usual meaning in triangle ABC]

A point P(x,y) moves so that the sum of the distance from P to the coordinate axes is equal to the distance from P to the point A(1,1). The equation of the locus of P in the first quadrant is -

Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that $\dfrac{QA}{AR} = \dfrac{RB}{BP} = \dfrac{PC}{CQ} = \dfrac{1}{2}$. Then $\dfrac{\text{Area}(\Delta PQR)}{\text{Area}(\Delta ABC)}$ is equal to

Let a point $A$ lie between the parallel lines $L_1$ and $L_2$ such that its distances from $L_1$ and $L_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle $ABC$, where the points $B$ and $C$ lie on the lines $L_1$ and $L_2$, respectively, is:

Let $ABC$ be an equilateral triangle with orthocentre at the origin and the side $BC$ on the line $x+2\sqrt{2}y=4$. If the co-ordinates of the vertex $A$ are $(\alpha,\beta)$, then the greatest integer less than or equal to $|\alpha+\sqrt{2}\beta|$ is

Suppose that a ray of light leaves the point $(3, 4)$, reflects off the $y$-axis towards the $x$-axis, reflects off the $x$-axis, and finally arrives at the point $(8, 2)$. The value of $x$ is:

Two vertices of a triangle are $(5, -1)$ and $(-2, 3)$. If orthocenter of the triangle is origin, then the co-ordinates of third vertex is: