Straight Lines and Pair of Straight Lines · Mathematics · JEE Advanced

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 ...

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...

Consider three points $$P = ( - \sin (\beta - \alpha ), - cos\beta ),Q = (cos(\beta - \alpha ),\sin \beta )$$ and $$R = (\cos (\beta - \alpha + \t...

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...

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 :

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

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 ...

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

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...

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

The orthocentre of the triangle formed by the lines $$xy=0$$ and $$x+y=1$$ 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

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...

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...

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

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 :

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. ...

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

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} &...

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 $$QR, RP, PQ$$...

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\...

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...

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 $$...

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...

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 ...

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,...

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 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 end $$A, B$$ of a straight line segment of constant length $$c$$ slide upon the fixed rectangular axes $$OX, OY$$ respectively. If the rectangle $...

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 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...

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...

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...

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 $$...

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 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 .................

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 $$2x + y + 5 = 0.$$