Mathematics
Straight Lines and Pair of Straight Lines
Previous Years Questions

## Numerical

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 ...
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 ...
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 $$x - y ... ## MCQ (Single Correct Answer) For$$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$$... A straight line$$L$$through the point$$(3, -2)$$is inclined at an angle$${60^ \circ }$$to the line$$\sqrt {3x} + y = 1.$$If$$L$$also inters... Let$$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$$inside the triangl... The lines$${L_1}:y - x = 0$$and$${L_2}:2x + y = 0$$intersect the line$${L_3}:y + 2 = 0$$at$$P$$and$$Q$$respectively. The bisector of the acu... Area of the triangle formed by the line$$x + y = 3$$and angle bisectors of the pair of straight line$${x^2} - {y^2} + 2y = 1$$is The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices$$\l...
Orthocentre of triangle with vertices $$\left( {0,0} \right),\left( {3,4} \right)$$ and $$\left( {4,0} \right)$$ is
A triangle with vertices $$(4, 0), (-1, -1), (3, 5)$$is
Locus of mid point of the portion between the axes of $$x$$ $$\cos \alpha + y\sin \alpha = p$$ where $$p$$ is constant is
If the pair of lines $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$ intersect on the $$y$$ axis then
The pair of lines represented by $$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$$ are perpendicular to each other for
Let $$0 < \alpha < {\pi \over 2}$$ be fixed angle. If $$P = \left( {\cos \theta ,\,\sin \theta } \right)$$ and $$Q = \left( {\cos \left( {\alp... Let$$P = \left( { - 1,\,0} \right),\,Q = \left( {0,\,0} \right)$$and$$R = \left( {3,\,3\sqrt 3 } \right)$$be three points. Then the equation of t... A straight line through the origin$$O$$meets the parallel lines$$4x+2y=9$$and$$2x+y+6=0$$at points$$P$$and$$Q$$respectively. Then the point ... The number of integer values of$$m$$, for which the$$x$$-coordinate of the point of intersection of the lines$$3x + 4y = 9$$and$$y = mx + 1$$is... Area of the parallelogram formed by the lines$$y = mx$$,$$y = mx + 1$$,$$y = nx$$and$$y = nx + 1$$equals Let$$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)$$... The incentre of the triangle with vertices$$\left( {1,\,\sqrt 3 } \right),\left( {0,\,0} \right)$$and$$\left( {2,\,0} \right)$$is Lt$$PQR$$be a right angled isosceles triangle, right angled at$$P(2, 1)$$. If the equation of the line$$QR$$is$$2x + y = 3,$$then the equation ... If$${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}} \r...
The diagonals of a parralleogram $$PQRS$$ are along the lines $$x + 3y = 4$$ and $$6x - 2y = 7$$. Then $$PQRS$$ must be a.
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 parr...
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
If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is
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 ...
If $$P=(1, 0),$$ $$Q=(-1, 0)$$ and $$R=(2, 0)$$ are three given points, then locus of the point $$S$$ satisfying the relation $$S{Q^2} + S{R^2} = 2S{P... The points$$\left( {0,{8 \over 3}} \right),\,\,\left( {1,\,3} \right)$$and$$\left( {82,\,30} \right)$$are vertices of A vector$$\overline a $$has components$$2p$$and$$1$$with respect to a rectangular cartesian system. This system is rotated through a certain ang... The straight lines$$x + y = 0,\,3x + y - 4 = 0,\,x + 3y - 4 = 0$$form a triangle which is The point$$\,\left( {4,\,1} \right)$$undergoes the following three transformations successively. Reflection about the line$$y=x$$. Translation thro... The points$$\left( { - a,\, - b} \right),\,\left( {0,\,0} \right),\,\left( {a,\,b} \right)$$and$$\left( {{a^2},\,ab} \right)$$are : ## Subjective The area of the triangle formed by intersection of a line parallel to$$x$$-axis and passing through$$P (h, k)$$with the lines$$y = x $$and$$x + ...
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...
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 th...
Let $$a, b, c$$ be real numbers with $${a^2} + {b^2} + {c^2} = 1.$$ Show that the equation $$\left| {\matrix{ {ax - by - c} & {bx + ay} &... For points$$P\,\,\, = \left( {{x_1},\,{y_1}} \right)$$and$$Q\,\,\, = \left( {{x_2},\,{y_2}} \right)$$of the co-ordinate plane, a new distance$$d\...
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 $$QR, RP, PQ$$...
Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.
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 $$x = -b,$$...
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 $$D$$ respectiv...
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 ...
Determine all values of $$\alpha$$ for which the point $$\left( {\alpha ,\,{\alpha ^2}} \right)$$ lies insides the triangle formed by the lines $$\... Show that all chords of the curve$$3{x^2} - {y^2} - 2x + 4y = 0,$$which subtend a right angle at the origin, pass through a fixed point. Find the co... Find the equation of the line passing through the point$$(2, 3)$$and making intercept of length 2 units between the lines$$y + 2x = 3$$and$$y + 2...
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 $$AB... A line cuts the$$x$$-axis at$$A (7, 0)$$and the$$y$$-axis at$$B (0, -5)$$. A variable line$$PQ$$is drawn perpendicular to$$AB$$cutting the$$...
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 $$F... Lines$${L_1} = ax + by + c = 0$$and$${L_2} = lx + my + n = 0$$intersect at the point$$P$$and make an angle$$\theta $$with each other. Find the ... One of the diameters of the circle circumscribing the rectangle$$ABCD$$is$$4y = x + 7$$. If$$A$$and$$B$$are the points$$(-3, 4)$$and$$(5, 4)...
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 $$(1,... Two equal sides of an isosceles triangle are given by the equations$$7x - y + 3 = 0$$and$$x + y - 3 = 0$$and its thirds side passes through the po... The vertices of a triangle are$$\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}} \rig...
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 end $$A, B$$ of a straight line segment of constant length $$c$$ slide upon the fixed rectangular axes $$OX, OY$$ respectively. If the rectangle \$...
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 $$5$$....
(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 po...
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 lin...
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 $$... One side of rectangle lies along the line$$4x + 7y + 5 = 0.$$Two of its vertices are$$(-3, 1)$$and$$(1, 1).$$Find the equations of the other thr... ## MCQ (More than One Correct Answer) Let$${L_1}$$be a straight line passing through the origin and$${L_2}$$be the straight line$$x + y = 1$$. If the intercepts made by the circle$${...
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 ...
All points lying inside the triangle formed by the points $$\left( {1,\,3} \right),\,\left( {5,\,0} \right)$$ and $$\left( { - 1,\,2} \right)$$ satisf...
Three lines $$px + qy + r = 0$$, $$qx + ry + p = 0$$ and $$rx + py + q = 0$$ are concurrent if

## Fill in the Blanks

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...
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...
The orthocentre of the triangle formed by the lines $$x + y = 1,\,2x + 3y = 6$$ and $$4x - y + 4 = 0$$ lies in quadrant number .............
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 .............
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 tha...
$$y = {10^x}$$ 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 .................

## True or False

The lines $$2x + 3y + 19 = 0$$ and $$9x + 6y - 17 = 0$$ cut the coordinates axes in concyclic points.
The straight line $$5x + 4y = 0$$ passes through the point of intersection of the straight lines $$x + 2y - 10 = 0$$ and $$2x + y + 5 = 0.$$
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