Straight Lines and Pair of Straight Lines

Let S denote the locus of the point of intersection of the pair of lines$4x - 3y = 12\alpha$,$4\alpha x + 3\alpha y = 12$,where $\alpha$ varies over the set of non-zero real numbers. Let T be the tangent to S passing through the points $(p,...

Consider the lines L1 and L2 defined by \({L_1}:x\sqrt 2 + y - 1 = 0\) and \({L_2}:x\sqrt 2 - y + 1 = 0\)For a fixed constant \(\lambda\), let C be the locus of a point P such that the product of the distance of P from L1 and the distance...

Consider the lines L1 and L2 defined by \({L_1}:x\sqrt 2 + y - 1 = 0\) and \({L_2}:x\sqrt 2 - y + 1 = 0\)For a fixed constant \(\lambda\), let C be the locus of a point P such that the product of the distance of P from L1 and the distance...

For a point \(P\) in the plane, Let \({d_1}\left( P \right)\) and \({d_2}\left( P \right)\) be the distance of the point \(P\) from the lines \(x - y = 0\) and \(x + y = 0\) respectively. The area of the region \(R\) consisting of all point...

For \(a > b > c > 0,\) the distance between \((1, 1)\) and the point of intersection of the lines \(ax + by + c = 0\) and \(bx + ay + c = 0\) is less than \(\left( {2\sqrt 2 } \right)\). Then

A straight line \(L\) through the point \((3, -2)\) is inclined at an angle \({60^ \circ }\) to the line \(\sqrt {3x} + y = 1.\) If \(L\) also intersects the x-axis, then the equation of \(L\) is

Consider the lines given by:
\({L_1}:x + 3y - 5 = 0\)
\({L_2}:3x - ky - 1 = 0\)
\({L_3}:5x + 2y - 12 = 0\)
Match the Statement/Expressions in Column I with the Statements/Expressions in Column II.

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Consider three points \(P = ( - \sin (\beta - \alpha ), - cos\beta ),Q = (cos(\beta - \alpha ),\sin \beta )\) and \(R = (\cos (\beta - \alpha + \theta ),\sin (\beta - \theta ))\) where \(0 < \alpha ,\beta ,\theta < {\pi \over 4}\). T...

A straight line through the vertex p of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then :

Let a and b be non-zero real numbers. Then, the equation \((a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0\) represents :

Lines \(\mathrm{L}_{1}: y-x=0\) and \(\mathrm{L}_{2}: 2 x+y=0\) intersect the line \(\mathrm{L}_{3}: y+2=0\) at \(\mathrm{P}\) and \(\mathrm{Q}\), respectively. The bisector of the acute angle between \(L_{1}\) and \(L_{2}\) intersects $$L_...

Let \(\mathrm{O(0,0), P(3,4), Q(6,0)}\) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

Consider the following linear equations
\(ax + by + cz = 0\)
\(bx + cy + az = 0\)
\(cx + ay + bz = 0\)
Match the conditions/expressions in Column I with statements in Column II.

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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 ...

Let \(O\left( {0,0} \right),P\left( {3,4} \right),Q\left( {6,0} \right)\) be the vertices of the triangles \(OPQ\). The point \(R\) inside the triangle \(OPQ\) is such that the triangles \(OPR\), \(PQR\), \(OQR\) are of equal area. The coor...

The area of the triangle formed by the intersection of a line parallel to X-axis and passing through \((h, k)\) with the lines \(y=x\) and \(x+y=2\) is \(4 h^{2}\). Find the locus of point \(P\).

The area of the triangle formed by intersection of a line parallel to \(x\)-axis and passing through \(P (h, k)\) with the lines \(y = x\) and \(x + y = 2\) is \(4{h^2}\). Find the locus of the point \(P\).

Area of the triangle formed by the line \(x + y = 3\) and angle bisectors of the pair of straight line \({x^2} - {y^2} + 2y = 1\) is

The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices \(\left( {0,0} \right),\left( {0,21} \right)\) and \(\left( {21,0} \right)\), is

Orthocentre of triangle with vertices \(\left( {0,0} \right),\left( {3,4} \right)\) and \(\left( {4,0} \right)\) is

Let \(0 < \alpha < {\pi \over 2}\) be fixed angle. If \(P = \left( {\cos \theta ,\,\sin \theta } \right)\) and \(Q = \left( {\cos \left( {\alpha - \theta } \right),\,\sin \left( {\alpha - \theta } \right)} \right),\) then \(Q\) is obtai...

A straight line through the origin \(O\) meets the parallel lines \(4x+2y=9\) and \(2x+y+6=0\) at points \(P\) and \(Q\) respectively. Then the point \(O\) divides the segemnt \(PQ\) in the ratio

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

A straight line \(L\) with negative slope passes through the point \((8, 2)\) and cuts the positive coordinate axes at points \(P\) and \(Q\). Find the absolute minimum value of \(OP + OQ,\) as \(L\) varies, where \(O\) is the origin.

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

A triangle with vertices \((4, 0), (-1, -1), (3, 5)\)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

A straight line \(L\) through the origin meets the lines \(x + y = 1\) and \(x + y = 3\) at \(P\) and \(Q\) respectively. Through \(P\) and \(Q\) two straight lines \({L_1}\) and \({L_2}\) are drawn, parallel to \(2x - y = 5\) and $$3x + y...

The number of integer values of \(m\), for which the \(x\)-coordinate of the point of intersection of the lines \(3x + 4y = 9\) and \(y = mx + 1\) is also an integer, is

Area of the parallelogram formed by the lines \(y = mx\), \(y = mx + 1\), \(y = nx\) and \(y = nx + 1\) equals

Let \(a, b, c\) be real numbers with \({a^2} + {b^2} + {c^2} = 1.\) Show that
the equation $$\left| {\matrix{
{ax - by - c} & {bx + ay} & {cx + a} \cr
{bx + ay} & { - ax + by - c} & {cy + b} \cr
{cx + a} & {cy + b} & { - ax -...

Let \(PS\) be the median of the triangle with vertices \(P(2, 2),\) \(Q(6, -1)\) and \(R(7, 3).\) The equation of the line passing through \((1, -1)\) and parallel to \(PS\) is

The incentre of the triangle with vertices \(\left( {1,\,\sqrt 3 } \right),\left( {0,\,0} \right)\) and \(\left( {2,\,0} \right)\) is

Let \(ABC\) and \(PQR\) be any two triangles in the same plane. Assume that the prependiculars from the points \(A, B, C\) to the sides \(QR, RP, PQ\) respectively are concurrent. Using vector methods or otherwise, prove that the prependicu...

For points \(P\,\,\, = \left( {{x_1},\,{y_1}} \right)\) and \(Q\,\,\, = \left( {{x_2},\,{y_2}} \right)\) of the co-ordinate plane, a new distance \(d\left( {P,\,Q} \right)\) is defined by \(d\left( {P,\,Q} \right)\)$$ = \left( {{x_2},\,{y_2...

Lt \(PQR\) be a right angled isosceles triangle, right angled at \(P(2, 1)\). If the equation of the line \(QR\) is \(2x + y = 3,\) then the equation representing the pair of lines \(PQ\) and \(PR\) is

Let \({L_1}\) be a straight line passing through the origin and \({L_2}\) be the straight line \(x + y = 1\). If the intercepts made by the circle \({x^2} + {y^2} - x + 3y = 0\) on \({L_1}\) and \({L_2}\) are equal, then which of the follow...

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

The diagonals of a parralleogram \(PQRS\) are along the lines \(x + 3y = 4\) and \(6x - 2y = 7\). Then \(PQRS\) must be a.

If the vertices \(P, Q, R\) of a triangle \(PQR\) are rational points, which of the following points of the triangle \(PQR\) is (are) always rational point(s)?

Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.

If \(\left( {P\left( {1,2} \right),\,Q\left( {4,6} \right),\,R\left( {5,7} \right)} \right)\) and \(S\left( {a,b} \right)\) are the vertices of a parrallelogram \(PQRS,\) then

A rectangle \(PQRS\) has its side \(PQ\) parallel to the line \(y = mx\) and vertices \(P, Q\) and \(S\) on the lines \(y = a, x = b\) and \(x = -b,\) respectively. Find the locus of the vertex \(R\).

The orthocentre of the triangle formed by the lines \(xy=0\) and \(x+y=1\) is

The locus of a variable point whose distance from \(\left( { - 2,\,0} \right)\) is \(2/3\) times its distance from the line \(x = - {9 \over 2}\) is

The equations to a pair of opposites sides of parallelogram are \({x^2} - 5x + 6 = 0\) and \({y^2} - 6y + 5 = 0,\) the equations to its diagonals are

Tagent at a point \({P_1}\) {other than \((0, 0)\)} on the curve \(y = {x^3}\) meets the curve again at \({P_2}\). The tangent at \({P_2}\) meets the curve at \({P_3}\), and so on. Show that the abscissae of $${P_1},\,{P_2},{P_3}......{P_n}...

The vertices of a triangle are \(A\left( { - 1, - 7} \right)B\left( {5,\,1} \right)\) and \(C\left( {1,\,4} \right).\) The equation of the bisector of the angle \(\angle ABC\) is ............... .

A line through \(A (-5, -4)\) meets the line \(x + 3y + 2 = 0,\) \(2x + y + 4 = 0\) and \(x - y - 5 = 0\) at the points \(B, C\) and \(D\) respectively. If $${\left( {15/AB} \right)^2} + {\left( {10/AC} \right)^2} = {\left( {6/AD} \right)...