Straight Lines and Pair of Straight Lines

If the area of the triangle formed by the lines $y=x+c$ and $2 x^2+5 x y+3 y^2=0$ is $\frac{1}{20}$ sq units, then $c=$

The diagonals $A C$ and $B D$ of a rhombus $A B C D$ intersect at the point $(3,4)$. If $B D=2 \sqrt{2}, A=(1,2), B=(\alpha, \beta)$, $C(5,6), D=(\gamma, \delta)$ and $\alpha<\delta<\gamma<\beta$, then $\beta+\gamma-\delta=$

Let $P=(-1,0), Q=(0,0)$ and $R=(3,3 \sqrt{3})$ be three Points. Then, the equation of the bisector of the $\angle P Q R$ is

If a line is moving between the coordinate axes such that the sum of the intercepts made by it on the coordinate axes is always 12, then the equation of that line which forms a triangle of maximum area with the coordinate axes is

If $2 x^2-3 x y+y^2=0$ represents two sides of a triangle and $x+y-1=0$ is its third side, then the distance between the orthocentre and the circumcentre of that triangle is

The lines $p\left(p^2+1\right) x-y+q=0$ and
$\left(p^2+1\right)^2 x+\left(p^2+1\right) y+2 q=0$ are perpendicular to a line $L$ for

Assertion (A) The difference of the slopes of the lines represented by $y^2-2 x y \sec ^2 \alpha+\left(3+\tan ^2 \alpha\right)$ $\left(-1+\tan ^2 \alpha\right) x^2=0$ is 4
Reason (R) The difference of the slopes represented by $a x^2+2 h x ...

The length of intercept of \(x+1=0\) between the lines \(3 x+2 y=5\) and \(3 x+2 y=3\) is

Suppose \(P\) and \(Q\) lie on \(3 x+4 y-4=0\) and \(5 x-y-4=0\) respectively. If the mid-point of \(P Q\) is \((1,5)\), then the slope of the line passing through \(P\) and \(Q\) is

Suppose that the sides passing through the vertex \((\alpha, \beta)\) of a triangle are bisected at right angles by the lines \(y^2-8 x y-9 x^2=0\). Then, the centroid of the triangle is

Suppose the slopes \(m_1\) and \(m_2\) of the lines represented by \(a x^2+2 h x y+b y^2=0\) satisfy \(3\left(m_1-m_2\right)-7=0\) and \(m_1 m_2-2=0\). Then, which of the following is true?

Starting from the point \(A(-3,4)\), a moving object touches \(2 x+y-7=0\) at \(B\) and reaches the point \(C(0,1)\). If the object travels along the shortest path, the distance between \(A\) and \(B\) is

Suppose a triangle is formed by \(x+y=10\) and the coordinate axes. Then, the number of points \((x, y)\) where \(x\) and \(y\) are natural numbers, lying inside the triangle is

The least distance from origin to a point on the line \(y=x+3\) which lies at a distance of 2 units from \((0,3)\) is

For \(\alpha \in\left[0, \frac{\pi}{2}\right]\), the angle between the lines represented by \([x \cos \theta-y] [(\cos \theta+\tan \alpha) x-(1-\cos \theta \tan \alpha) y]=0\) is

If the lines represented by \(a x^2+2 h x y+b y^2+2 g x+2 f y+c=0\) intersect on the \(X\)-axis, which of the following is in general incorrect?

If the points of intersection of the coordinate axes and \(|x+y|=2\) form a rhombus, then its area is

Suppose \(P\) and \(Q\) are the mid-points of the sides \(A B\) and \(B C\) of a triangle where \(A(1,3), B(3,7)\) and \(C(7,15)\) are vertices. Then, the locus of \(R\) satisfying \(A C^2+Q R^2=P R^2\) is

Suppose, in \(\triangle A B C, x-y+5=0, x+2 y=0\) are respectively the equations of the perpendicular bisectors of the sides \(A B\) and \(A C\). If \(A\) is \((1,-2)\), the equation of the line joining \(B\) and \(C\) is

A line passing through \(P(2,3)\) and making an angle of \(30^{\circ}\) with the positive direction of \(X\)-axis meets \(x^2-2 x y-y^2=0\) at \(A\) and \(B\). Then the value of \(P A: P B\) is

If the pair of straight lines \(9 x^2+a x y+4 y^2+6 x+b y-3=0\) represents two parallel lines, then

If the bisectors of the pair of lines \(x^2-2 m x y-y^2=0\) is represented by \(x^2-2 n x y-y^2=0\), then

The equation of bisector of the angle between the lines represented by \(3 x^2-5 x y+4 y^2=0\) is

If the lines, joining the origin to the points of intersection of the curve \(2 x^2-2 x y+3 y^2+2 x-y-1=0\) and the line \(x+2 y=k\), are at right angles, then \(k^2\) equals

The acute angle between lines \(6 x^2+11 x y-10 y^2=0\) is

The equation of a straight line which passes through the point \(\left(a \cos ^3 \theta, a \sin ^3 \theta\right)\) and perpendicular to \((x \sec \theta+y \operatorname{cosec} \theta)=a\) is

The equation of the base of an equilateral triangle is \(x+y=2\) and one vertex is \((2,-1)\), then the length of the side of the triangle is

Line has slope \(m\) and \(y\)-intercept 4 . The distance between the origin and the line is equal to

Find the equation of a straight line passing through \((-5,6)\) and cutting off equal intercepts on the coordinate axes.

When the axes are rotated through an angle
45\(^\circ\), the new coordinates of a point P are
(1, \(-\)1). The coordinates of P in the original
system are

If the lines represented by the equation \(2 x^2-p x y+2 y^2=0\) are real, then the value of \(p\) lies in the interval

If one of the line represented by \(-a x^2+2 h x y+b y^2=0\) passes through \((2,3)\) and the other passes through \((4,5)\), then \(a+2 h+b\) equals

The value of \(p\) for which the equation \(x^2+p x y+y^2-5 x-7 y+6=0\) represents a pair of straight lines is

Find the equation of a line which passes through \(\left(2 \cos ^3(\theta), 2 \sin ^3(\theta)\right)\) and is perpendicular to the line \(x \cos (\theta)-y \sin (\theta)=2 \cos (2 \theta)\).

The distance of the point \((1,2)\) from the line \(x+y+5=0\) measured along the line parallel to \(3 x-y=7\) is equal to

\(C\) is the centroid of the triangle with vertices \((3,-1),(1,3)\) and \((2,4)\). Let \(P\) be the point of intersection of the lines \(x+3 y-1=0\) and \(3 x-y+1=0\). Then a line which passes through both points \(C\) and \(P\) would also...

If \(A(1,3)\) and \(C(7,5)\) are two opposite vertices of a square, then find the equation of a side passing through \(A\).

The point to which the origin should be shifted in order to eliminate the \(x\) and \(y\) terms from the equation \(9 x^2+4 y^2+10 x+12 y+1=0\) is

If the slope of one of the lines represented by \(a x^2+2 h x y+b y^2=0\) is the square of the other then, \(\left|\frac{a+b}{h}+\frac{8 h^2}{a b}\right|\) is equal to

If \(m_1\) and \(m_2,\left(m_1>m_2\right)\) are the slopes of the lines represented by \(5 x^2-8 x y+3 y^2=0\), then \(m_1: m_2\) equals

If the centroid of the triangle formed by the lines \(2 y^2+5 x y-3 x^2=0\) and \(x+y=k\) is \(\left(\frac{1}{18}, \frac{11}{18}\right)\), then the value of \(k\) equals

The equation of the pair of straight lines perpendicular to the pair \(2 x^2+3 x y+2 y^2+10 x+5 y=0\) and passing though the origin is

If \(2x+3y+4=0\) is the perpendicular bisector of the line segment joining the points A(1, 2) and B(\(\alpha,\beta\)), then the value of \(13\alpha+13\beta\) equals

For three consecutive odd integers \(a \cdot b\) and \(c\), if the variable line \(a x+b y+c=0\) always passes through the point \((\alpha, \beta)\), the value of \(\alpha^2+\beta^2\) equals

the sum of the squares of the intercepts made the line \(5x-2y=10\) on the coordinate axes equals

If the axes are rotated through an angle \(45 \Upsilon\), the coordinates of the point \((2 \sqrt{2},-3 \sqrt{2})\) in the new system are

The angle between the lines represented by \(\cos \theta(\cos \theta+1) x^2-\left(2 \cos \theta+\sin ^2 \theta\right) x y+(1-\cos \theta) y^2=0\) is

If the acute angle between lines \(a x^2+2 h x y+b y^2=0\) is \(\frac{\pi}{4}\), then \(4 h^2\) is equal to

Find the angle between the pair of lines represented by the equation \(x^2+4 x y+y^2=0\).

The equation \(8 x^2-24 x y+18 y^2-6 x+9 y-5=0\) represents a