Straight Lines and Pair of Straight Lines

If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is

Determine all values of \(\alpha\) for which the point \(\left( {\alpha ,\,{\alpha ^2}} \right)\) lies insides the triangle formed by the lines
$$$\matrix{
{2x + 3y - 1 = 0} \cr
{x + 2y - 3 = 0} \cr
{5x - 6y - 1 = 0} \cr

...

Let the algebraic sum of the perpendicular distances from the points \(\left( {2,0} \right),\,\left( {0,\,2} \right)\) \(\left( {1,\,1} \right)\) to a variable straight line be zero; then the line passes through a fixed point whose cordinat...

Show that all chords of the curve \(3{x^2} - {y^2} - 2x + 4y = 0,\) which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

Find the equation of the line passing through the point \((2, 3)\) and making intercept of length 2 units between the lines \(y + 2x = 3\) and \(y + 2x = 5\).

A line cuts the \(x\)-axis at \(A (7, 0)\) and the \(y\)-axis at \(B (0, -5)\). A variable line \(PQ\) is drawn perpendicular to \(AB\) cutting the \(x\)axis in \(P\) and they \(Y\)-axis in \(Q\). If \(AQ\) and \(BP\) intersect at \(R\), fi...

Line \(L\) has intercepts \(a\) and \(b\) on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line \(L\) has intercepts \(p\) and \(q\), then

Straight lines \(3x + 4y = 5\) and \(4x - 3y = 15\) intersect at the point \(A\). Points \(B\) and \(C\) are choosen on these two lines such that \(AB = AC\). Determine the possible equations of the line \(BC\) passing through the point $$(...

Let \(ABC\) be a triangle with \(AB = AC\). If \(D\) is the midpoint of \(BC, E\) is the foot of the perpendicular drawn from \(D\) to \(AC\) and \(F\) the mid-point of \(DE\), prove that \(AF\) is perpendicular to \(BE\).

Lines\({L_1} = ax + by + c = 0\) and \({L_2} = lx + my + n = 0\) intersect at the point \(P\) and make an angle \(\theta\) with each other. Find the equation of a line \(L\) different from \({L_2}\) which passes through \(P\) and makes the...

If \(P=(1, 0),\) \(Q=(-1, 0)\) and \(R=(2, 0)\) are three given points, then locus of the point \(S\) satisfying the relation \(S{Q^2} + S{R^2} = 2S{P^2},\) is

The lines \(2x + 3y + 19 = 0\) and \(9x + 6y - 17 = 0\) cut the coordinates axes in concyclic points.

A vector \(\overline a\) has components \(2p\) and \(1\) with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, $$...

The points \(\left( {0,{8 \over 3}} \right),\,\,\left( {1,\,3} \right)\) and \(\left( {82,\,30} \right)\) are vertices of

All points lying inside the triangle formed by the points \(\left( {1,\,3} \right),\,\left( {5,\,0} \right)\) and \(\left( { - 1,\,2} \right)\) satisfy

The orthocentre of the triangle formed by the lines \(x + y = 1,\,2x + 3y = 6\) and \(4x - y + 4 = 0\) lies in quadrant number .............

Two sides of rhombus \(ABCD\) are parallel to the lines \(y = x + 2\) and \(y = 7x + 3\). If the diagonals of the rhombus intersect at the point \((1, 2)\) and the vertex \(A\) is on the \(y\)-axis, find possible co-ordinates of \(A\).

Three lines \(px + qy + r = 0\), \(qx + ry + p = 0\) and \(rx + py + q = 0\) are concurrent if

One of the diameters of the circle circumscribing the rectangle \(ABCD\) is \(4y = x + 7\). If \(A\) and \(B\) are the points \((-3, 4)\) and \((5, 4)\) respectively, then find the area of rectangle.

Two equal sides of an isosceles triangle are given by the equations \(7x - y + 3 = 0\) and \(x + y - 3 = 0\) and its thirds side passes through the point \((1, -10)\). Determine the equation of the third side.

If \(a,\,b\) and \(c\) are in A.P., then the straight line \(ax + by + c = 0\) will always pass through a fixed point whose coordinates are ...............

The straight line \(5x + 4y = 0\) passes through the point of intersection of the straight lines \(x + 2y - 10 = 0\) and \(2x + y + 5 = 0.\)

The end \(A, B\) of a straight line segment of constant length \(c\) slide upon the fixed rectangular axes \(OX, OY\) respectively. If the rectangle \(OAPB\) be completed, then show that the locus of the foot of the perpendicular drawn from...

The straight lines \(x + y = 0,\,3x + y - 4 = 0,\,x + 3y - 4 = 0\) form a triangle which is

Given the points \(A\left( {0,4} \right)\) and \(B\left( {0, - 4} \right)\), the equation of the locus of the point \(P\left( {x,y} \right)\) such that \(\left| {AP - BP} \right| = 6\) is .............

The vertices of a triangle are \(\left[ {a{t_1}{t_2},\,\,a\left( {{t_1} + {t_2}} \right)} \right],\,\,\left[ {a{t_2}{t_3},a\left( {{t_2} + {t_3}} \right)} \right],\,\,\left[ {a{t_3}{t_1},\,a\left( {{t_3} + {t_1}} \right)} \right]\). Find ...

The coordinates of \(A, B, C\) are \((6, 3), (-3, 5), (4, -2)\) respectively, and \(P\) is any point \((x, y)\). Show that the ratio of the area of the triangles \(\Delta\) \(PBC\) and \(\Delta\)\(ABC\) is $$\left| {{{x + y - 2} \over 7...

\(y = {10^x}\) is the reflection of \({\log _{10}}\,x\) in the line whose equation is ...........

The set of lines \(ax + by + c = 0,\) where \(3a + 2b + 4c = 0\) is concurrent at the point ..........

The area enclosed within the curve \(\left| x \right| + \left| y \right| = 1\) is .................

A straight line \(L\) is perpendicular to the line \(5x - y = 1.\) The area of the triangle formed by the line \(L\) and the coordinate axes is \(5\). Find the equation of the Line \(L\).

The point \(\,\left( {4,\,1} \right)\) undergoes the following three transformations successively.
Reflection about the line \(y=x\).
Translation through a distance 2 units along the positive direction of x-axis.
Rotation through an angle ...

(a) Two vertices of a triangle are \((5, -1)\) and \((-2, 3).\) If the orthocentre of the triangle is the origin, find the coordinates of the third point.
(b) Find the equation of the line which bisects the obtuse angle between the lines $...

The points \(\left( { - a,\, - b} \right),\,\left( {0,\,0} \right),\,\left( {a,\,b} \right)\) and \(\left( {{a^2},\,ab} \right)\) are :

A straight line segment of length \(\ell\) moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio \(1 : 2\)

One side of rectangle lies along the line \(4x + 7y + 5 = 0.\) Two of its vertices are \((-3, 1)\) and \((1, 1).\) Find the equations of the other three sides.

The area of a triangle is \(5\). Two of its vertices are \(A\left( {2,1} \right)\) and \(B\left( {3, - 2} \right)\). The third vertex \(C\) lies on \(y = x + 3\). Find \(C\).