Straight Lines Questions (433)

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

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).