Straight Lines and Pair of Straight Lines

A ray of light along \(x + \sqrt 3 y = \sqrt 3\) gets reflected upon reaching \(X\)-axis, the equation of the reflected ray is :

The \(x\)-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as \((0, 1) (1, 1)\) and \((1, 0)\) is :

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 :

The lines \({L_1}:y - x = 0\) and \({L_2}:2x + y = 0\) intersect the line \({L_3}:y + 2 = 0\) at \(P\) and \(Q\) respectively. The bisector of the acute angle between \({L_1}\) and \({L_2}\) intersects \({L_3}\) at \(R\).
Statement-1: The ...

The line \(L\) given by \({x \over 5} + {y \over b} = 1\) passes through the point \(\left( {13,32} \right)\). The line K is parrallel to \(L\) and has the equation \({x \over c} + {y \over 3} = 1.\) Then the distance between \(L\) and $$K$...

The shortest distance between the line \(y - x = 1\) and the curve \(x = {y^2}\) is :

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

The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of k is :

Let \(P = \left( { - 1,0} \right),\,Q = \left( {0,0} \right)\) and \(R = \left( {3,3\sqrt 3 } \right)\) be three point. The equation of the bisector of the angle \(PQR\) is :

If one of the lines of \(m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0\) is a bisector of angle between the lines \(xy = 0,\) then \(m\) is :

Let A \(\left( {h,k} \right)\), B\(\left( {1,1} \right)\) and C \((2, 1)\) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is \(1\) square unit, then the set of values which \('k'\) can take...

A straight line through the point \(A (3, 4)\) is such that its intercept between the axes is bisected at \(A\). Its equation is :

If \(\left( {a,{a^2}} \right)\) falls inside the angle made by the lines \(y = {x \over 2},\) \(x > 0\) and \(y = 3x,\) \(x > 0,\) then a belong to :

The line parallel to the \(x\) - axis and passing through the intersection of the lines \(ax + 2by + 3b = 0\) and \(bx - 2ay - 3a = 0,\) where \((a, b)\) \(\ne\) \((0, 0)\) is :

If non zero numbers \(a, b, c\) are in \(H.P.,\) then the straight line \({x \over a} + {y \over b} + {1 \over c} = 0\) always passes through a fixed point. That point is :

If a vertex of a triangle is \((1, 1)\) and the mid points of two sides through this vertex are \((-1, 2)\) and \((3, 2)\) then the centroid of the triangle is :

Let \(A\left( {2, - 3} \right)\) and \(B\left( {-2, 1} \right)\) be vertices of a triangle \(ABC\). If the centroid of this triangle moves on the line \(2x + 3y = 1\), then the locus of the vertex \(C\) is the line :

The equation of the straight line passing through the point \((4, 3)\) and making intercepts on the co-ordinate axes whose sum is \(-1\) is :

If the sum of the slopes of the lines given by \({x^2} - 2cxy - 7{y^2} = 0\) is four times their product \(c\) has the value :

If one of the lines given by \(6{x^2} - xy + 4c{y^2} = 0\) is \(3x + 4y = 0,\) then \(c\) equals :

If the pair of straight lines \({x^2} - 2pxy - {y^2} = 0\) and \({x^2} - 2qxy - {y^2} = 0\) be such that each pair bisects the angle between the other pair, then :

If the equation of the locus of a point equidistant from the point \(\left( {{a_{1,}}{b_1}} \right)\) and \(\left( {{a_{2,}}{b_2}} \right)\) is
\(\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0\) , then the valu...

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 \(\alpha \left( {0 < \alpha < {\pi \over 4}} \right)\) with the positive direction of x-axis. The equation o...

Locus of centroid of the triangle whose vertices are \(\left( {a\cos t,a\sin t} \right),\left( {b\sin t, - b\cos t} \right)\) and \(\left( {1,0} \right),\) where \(t\) is a parameter, is :

If \({x_1},{x_2},{x_3}\) and \({y_1},{y_2},{y_3}\) are both in G.P. with the same common ratio, then the points \(\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)\) and \(\left( {{x_3},{y_3}} \right)\) :

A triangle with vertices \(\left( {4,0} \right),\left( { - 1, - 1} \right),\left( {3,5} \right)\) is :

The pair of lines represented by
\($3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0\)$
are perpendicular to each other for :

If the pair of lines
\(a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0\)
intersect on the \(y\)-axis then :

Locus of mid point of the portion between the axes of
\(x\) \(cos\) \(\alpha + y\,\sin \alpha = p\) where \(p\) is constant is :