Straight Lines and Pair of Straight Lines
563 Questions
2002
JEE Advanced
MCQ
IIT-JEE 2002
A triangle with vertices $(4, 0), (-1, -1), (3, 5)$is
A.
isosceles and right angled
B.
isosceles but not right angled
C.
right angled but not isosceles
D.
neither right angled nor isosceles
2002
JEE Advanced
MCQ
IIT-JEE 2002
Locus of mid point of the portion between the axes of $x$ $\cos \alpha + y\sin \alpha = p$ where $p$ is constant is
A.
${x^2} + {y^2} = {4 \over {{p^2}}}\,\,\,$
B.
${x^2} + {y^2} = 4{p^2}$
C.
${1 \over {{x^2}}} + {1 \over {{y^2}}} = {2 \over {{p^2}}}$
D.
${1 \over {{x^2}}} + {1 \over {{y^2}}} = {4 \over {{p^2}}}$
2002
JEE Advanced
MCQ
IIT-JEE 2002
The pair of lines represented by
$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$ are perpendicular to each other for
$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$ are perpendicular to each other for
A.
two values of $a$
B.
$\forall \,a$
C.
for one values of $a$
D.
for no values of $a$
2002
JEE Advanced
Numerical
IIT-JEE 2002
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 $3x + y = 5$ respectively. Lines ${L_1}$ and ${L_2}$ intersect at $R$. Show that the locus of $R$, as $L$ varies is a straight line.
Correct Answer: Solve it.
2002
JEE Advanced
Numerical
IIT-JEE 2002
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.
Correct Answer: 18
2001
JEE Advanced
MCQ
IIT-JEE 2001 Screening
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
A.
2
B.
0
C.
4
D.
1
2001
JEE Advanced
MCQ
IIT-JEE 2001 Screening
Area of the parallelogram formed by the lines $y = mx$, $y = mx + 1$, $y = nx$ and $y = nx + 1$ equals
A.
$\left| {m + n} \right|/{\left( {m - n} \right)^2}$
B.
$2/\left| {m + n} \right|$
C.
$1/\left( {\left| {m + n} \right|} \right)$
D.
$1/\left( {\left| {m - n} \right|} \right)$
2001
JEE Advanced
Numerical
IIT-JEE 2001
Let $a, b, c$ be real numbers with ${a^2} + {b^2} + {c^2} = 1.$ Show that
represents a straight line.
the equation $\left| {\matrix{ {ax - by - c} & {bx + ay} & {cx + a} \cr {bx + ay} & { - ax + by - c} & {cy + b} \cr {cx + a} & {cy + b} & { - ax - by + c} \cr } } \right| = 0$
represents a straight line.
Correct Answer: Solve it.
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
The incentre of the triangle with vertices $\left( {1,\,\sqrt 3 } \right),\left( {0,\,0} \right)$ and $\left( {2,\,0} \right)$ is
A.
$\left( {1,\,{{\sqrt 3 } \over 2}} \right)$
B.
$\left( {{2 \over 3},\,{1 \over {\sqrt 3 }}} \right)$
C.
$\left( {{2 \over 3},\,{{\sqrt 3 } \over 2}} \right)$
D.
$\left( {1,\,{1 \over {\sqrt 3 }}} \right)$
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
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
A.
$2x - 9y - 7 = 0$
B.
$2x - 9y - 11 = 0$
C.
$2x + 9y - 11 = 0$
D.
$2x + 9y + 7 = 0$
2000
JEE Advanced
Numerical
IIT-JEE 2000
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}} \right)\left| {{x_1} - {x_2}} \right| + \left| {{y_1} - {y_2}} \right|.$ Let $O = (0, 0)$ and $A = (3, 2)$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $O$ and $A$ consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.
Correct Answer: Solve it.
2000
JEE Advanced
Numerical
IIT-JEE 2000
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 prependiculars from $P, Q, R $ to $BC,$ $CA$, $AB$ respectively are also concurrent.
Correct Answer: Solve it.
1999
JEE Advanced
MCQ
IIT-JEE 1999
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).$
A.
lie on a straight line
B.
lie on an ellipse
C.
lie on a circle
D.
are vertices of a triangle
1999
JEE Advanced
MCQ
IIT-JEE 1999
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
A.
$3{x^2} - 3{y^2} + 8xy + 20x + 10y + 25 = 0$
B.
$3{x^2} - 3{y^2} + 8xy - 20x - 10y + 25 = 0$
C.
$3{x^2} - 3{y^2} + 8xy + 10x + 15y + 20 = 0$
D.
$3{x^2} - 3{y^2} - 8xy - 10x - 15y - 20 = 0$
1999
JEE Advanced
MSQ
IIT-JEE 1999
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 following equations can represent ${L_1}$?
A.
$x + y = 0$
B.
$x -y = 0$
C.
$x + 7y = 0$
D.
$x - 7y = 0$
1998
JEE Advanced
MCQ
IIT-JEE 1998
The diagonals of a parralleogram $PQRS$ are along the lines $x + 3y = 4$ and $6x - 2y = 7$. Then $PQRS$ must be a.
A.
rectangle
B.
square
C.
cyclic quadrilateral
D.
rhombus.
1998
JEE Advanced
MCQ
IIT-JEE 1998
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.
$a = 2,\,b = 4$
B.
$a = 3,\,b = 4$
C.
$a = 2,\,b = 3$
D.
$a = 3,\,b = 5$
1998
JEE Advanced
MSQ
IIT-JEE 1998
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)?
A.
centroid ( A rational point is a point both of whose co-ordinates are rational numbers.)
B.
incentre. ( A rational point is a point both of whose co-ordinates are rational numbers.)
C.
circumcentre ( A rational point is a point both of whose co-ordinates are rational numbers.)
D.
orthocentre ( A rational point is a point both of whose co-ordinates are rational numbers.)
1998
JEE Advanced
Numerical
IIT-JEE 1998
Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.
Correct Answer: Solve it.
1996
JEE Advanced
Numerical
IIT-JEE 1996
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$.
Correct Answer: $$x\left( {{m^2} - 1} \right) - ym + \left( {{m^2} + 1} \right)b + am = 0$$
1995
JEE Advanced
MCQ
IIT-JEE 1995
The orthocentre of the triangle formed by the lines $xy=0$ and $x+y=1$ is
A.
$\left( {{1 \over 2},\,{1 \over 2}} \right)$
B.
$\left( {{1 \over 3},\,{1 \over 3}} \right)$
C.
$\left( {0,\,0} \right)$
D.
$\left( {{1 \over 4},\,{1 \over 4}} \right)$
1994
JEE Advanced
MCQ
IIT-JEE 1994
The locus of a variable point whose distance from $\left( { - 2,\,0} \right)$ is $2/3$ times its distance from the line $x = - {9 \over 2}$ is
A.
ellipse -
B.
parabola
C.
hyperbola
D.
none of these
1994
JEE Advanced
MCQ
IIT-JEE 1994
The equations to a pair of opposites sides of parallelogram are ${x^2} - 5x + 6 = 0$ and ${y^2} - 6y + 5 = 0,$ the equations to its diagonals are
A.
$x + 4y = 13,\,y = 4x - 7$
B.
$4x + y = 13,\,4y = x - 7$
C.
$4x + y = 13,\,y = 4x - 7$
D.
$y - 4x = 13,\,y + 4x = 7$
1993
JEE Advanced
Numerical
IIT-JEE 1993
A line through $A (-5, -4)$ meets the line $x + 3y + 2 = 0,$ $2x + y + 4 = 0$ and $x - y - 5 = 0$ at the points $B, C$ and $D$ respectively. If ${\left( {15/AB} \right)^2} + {\left( {10/AC} \right)^2} = {\left( {6/AD} \right)^2},$ find the equation of the line.
Correct Answer: $$2x + 3y + 22 = 0$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
Tagent at a point ${P_1}$ {other than $(0, 0)$} on the curve $y = {x^3}$ meets the curve again at ${P_2}$. The tangent at ${P_2}$ meets the curve at ${P_3}$, and so on. Show that the abscissae of ${P_1},\,{P_2},{P_3}......{P_n},$ form a G.P. Also find the ratio.
[area $\left( {\Delta {P_1},{P_2},{P_3}} \right)$]/[area $\left( {{P_2},{P_3},{P_4}} \right)$]
Correct Answer: $${1 \over {64}}\,sq.\,unit$$
1993
JEE Advanced
Numerical
IIT-JEE 1993
The vertices of a triangle are $A\left( { - 1, - 7} \right)B\left( {5,\,1} \right)$ and $C\left( {1,\,4} \right).$ The equation of the bisector of the angle $\angle ABC$ is ............... .
Correct Answer: $$x - 7y + 2 = 0$$
1992
JEE Advanced
MCQ
IIT-JEE 1992
If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is
A.
square
B.
circle
C.
straight line
D.
two intersecting lines
1992
JEE Advanced
Numerical
IIT-JEE 1992
Determine all values of $\alpha $ for which the point $\left( {\alpha ,\,{\alpha ^2}} \right)$ lies insides the triangle formed by the lines
$$\matrix{
{2x + 3y - 1 = 0} \cr
{x + 2y - 3 = 0} \cr
{5x - 6y - 1 = 0} \cr
} $$
Correct Answer: $$\alpha \in \left( { - {3 \over 2}, - 1} \right) \cup \left( {{1 \over 2},1} \right)$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
Find the equation of the line passing through the point $(2, 3)$ and making intercept of length 2 units between the lines $y + 2x = 3$ and $y + 2x = 5$.
Correct Answer: $$3x + 4y - 18 = 0$$ or $$x - 2 = 0$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
Show that all chords of the curve $3{x^2} - {y^2} - 2x + 4y = 0,$ which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.
Correct Answer: $$(1, -2)$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
Let the algebraic sum of the perpendicular distances from the points $\left( {2,0} \right),\,\left( {0,\,2} \right)$ $\left( {1,\,1} \right)$ to a variable straight line be zero; then the line passes through a fixed point whose cordinates are ...............
Correct Answer: $$\left( {1,\,1} \right)$$
1990
JEE Advanced
MCQ
IIT-JEE 1990
Line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line $L$ has intercepts $p$ and $q$, then
A.
${a^2} + {b^2} = {p^2} + {q^2}$
B.
${1 \over {{a^2}}} + {1 \over {{b^2}}} = {1 \over {{p^2}}} + {1 \over {{q^2}}}$
C.
${a^2} + {p^2} = {b^2} + {q^2}$
D.
${1 \over {{a^2}}} + {1 \over {{p^2}}} = {1 \over {{b^2}}} + {1 \over {{q^2}}}$
1990
JEE Advanced
Numerical
IIT-JEE 1990
Straight lines $3x + 4y = 5$ and $4x - 3y = 15$ intersect at the point $A$. Points $B$ and $C$ are choosen on these two lines such that $AB = AC$. Determine the possible equations of the line $BC$ passing through the point $(1, 2)$.
Correct Answer: $$x - 7y + 13 = 0$$ or $$7x + y - 9 = 0$$
1990
JEE Advanced
Numerical
IIT-JEE 1990
A line cuts the $x$-axis at $A (7, 0)$ and the $y$-axis at $B (0, -5)$. A variable line $PQ$ is drawn perpendicular to $AB$ cutting the $x$axis in $P$ and they $Y$-axis in $Q$. If $AQ$ and $BP$ intersect at $R$, find the locus of R.
Correct Answer: $${x^2} + {y^2} - 7x + 5y = 0$$
1989
JEE Advanced
Numerical
IIT-JEE 1989
Let $ABC$ be a triangle with $AB = AC$. If $D$ is the midpoint of $BC, E$ is the foot of the perpendicular drawn from $D$ to $AC$ and $F$ the mid-point of $DE$, prove that $AF$ is perpendicular to $BE$.
Correct Answer: Solve it.
1988
JEE Advanced
MCQ
IIT-JEE 1988
If $P=(1, 0),$ $Q=(-1, 0)$ and $R=(2, 0)$ are three given points, then locus of the point $S$ satisfying the relation $S{Q^2} + S{R^2} = 2S{P^2},$ is
A.
a straight line parallel to x-axis
B.
a circle passing through the origin
C.
a circle with the centre at the origin
D.
a straight line parallel to y-axis.
1988
JEE Advanced
Numerical
IIT-JEE 1988
Lines${L_1} = ax + by + c = 0$ and ${L_2} = lx + my + n = 0$ intersect at the point $P$ and make an angle $\theta $ with each other. Find the equation of a line $L$ different from ${L_2}$ which passes through $P$ and makes the same angle $\theta $ with ${L_1}$.
Correct Answer: $$\left( {{a^2} + {b^2}} \right)\left( {\ell x + my + n} \right) - 2\left( {a\ell + bm} \right)\left( {ax + by + c} \right) = 0$$
1988
JEE Advanced
MCQ
IIT-JEE 1988
The lines $2x + 3y + 19 = 0$ and $9x + 6y - 17 = 0$ cut the coordinates axes in concyclic points.
A.
TRUE
B.
FALSE
1986
JEE Advanced
MCQ
IIT-JEE 1986
The points $\left( {0,{8 \over 3}} \right),\,\,\left( {1,\,3} \right)$ and $\left( {82,\,30} \right)$ are vertices of
A.
an obtuse angled triangle
B.
an acute angled triangle
C.
a right angled triangle
D.
none of these
1986
JEE Advanced
MCQ
IIT-JEE 1986
A vector $\overline a $ has components $2p$ and $1$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, $\overline a $ has components $p + 1$ and $1$, then
A.
$p = 0$
B.
$p = 1$ or $p = - {1 \over 3}$
C.
$\,p = - 1$ or $p = {1 \over 3}$
D.
$p = 1$ or $p = -1$
1986
JEE Advanced
MSQ
IIT-JEE 1986
All points lying inside the triangle formed by the points $\left( {1,\,3} \right),\,\left( {5,\,0} \right)$ and $\left( { - 1,\,2} \right)$ satisfy
A.
$3x + 2y \ge 0$
B.
$2x + y - 13 \ge 0$
C.
$2x - 3y - 12 \le 0$
D.
$ - 2x + y \ge 0$
1985
JEE Advanced
MSQ
IIT-JEE 1985
Three lines $px + qy + r = 0$, $qx + ry + p = 0$ and $rx + py + q = 0$ are concurrent if
A.
$p + q + r = 0$
B.
${p^2} + {q^2} + {r^2} = qr + rp + pq$
C.
${p^3} + {q^3} + {r^3} = 3pqr$
D.
none of these.
1985
JEE Advanced
Numerical
IIT-JEE 1985
One of the diameters of the circle circumscribing the rectangle $ABCD$ is $4y = x + 7$. If $A$ and $B$ are the points $(-3, 4)$ and $(5, 4)$ respectively, then find the area of rectangle.
Correct Answer: 32 sq. units.
1985
JEE Advanced
Numerical
IIT-JEE 1985
Two sides of rhombus $ABCD$ are parallel to the lines $y = x + 2$ and $y = 7x + 3$. If the diagonals of the rhombus intersect at the point $(1, 2)$ and the vertex $A$ is on the $y$-axis, find possible co-ordinates of $A$.
Correct Answer: $$(0 ; 0)$$ or $$\left( {0,\,{\raise0.5ex\hbox{$\scriptstyle 5$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}} \right)$$
1985
JEE Advanced
Numerical
IIT-JEE 1985
The orthocentre of the triangle formed by the lines $x + y = 1,\,2x + 3y = 6$ and $4x - y + 4 = 0$ lies in quadrant number .............
Correct Answer: First
1984
JEE Advanced
Numerical
IIT-JEE 1984
Two equal sides of an isosceles triangle are given by the equations $7x - y + 3 = 0$ and $x + y - 3 = 0$ and its thirds side passes through the point $(1, -10)$. Determine the equation of the third side.
Correct Answer: $$x - 3y - 31 = 0$$ or $$3x + y + 7 = 0$$
1984
JEE Advanced
Numerical
IIT-JEE 1984
If $a,\,b$ and $c$ are in A.P., then the straight line $ax + by + c = 0$ will always pass through a fixed point whose coordinates are ...............
Correct Answer: $$\left( {1,\, - 2} \right)$$
1983
JEE Advanced
MCQ
IIT-JEE 1983
The straight lines $x + y = 0,\,3x + y - 4 = 0,\,x + 3y - 4 = 0$ form a triangle which is
A.
isosceles
B.
equilateral
C.
right angled
D.
none of these
1983
JEE Advanced
Numerical
IIT-JEE 1983
The vertices of a triangle are
$\left[ {a{t_1}{t_2},\,\,a\left( {{t_1} + {t_2}} \right)} \right],\,\,\left[ {a{t_2}{t_3},a\left( {{t_2} + {t_3}} \right)} \right],\,\,\left[ {a{t_3}{t_1},\,a\left( {{t_3} + {t_1}} \right)} \right]$. Find the orthocentre of the triangle.
$\left[ {a{t_1}{t_2},\,\,a\left( {{t_1} + {t_2}} \right)} \right],\,\,\left[ {a{t_2}{t_3},a\left( {{t_2} + {t_3}} \right)} \right],\,\,\left[ {a{t_3}{t_1},\,a\left( {{t_3} + {t_1}} \right)} \right]$. Find the orthocentre of the triangle.
Correct Answer: $$\left( { - a,a\left( {{t_1} + {t_2} + {t_3}} \right) + a{t_1}{t_2}{t_3}} \right)$$
1983
JEE Advanced
Numerical
IIT-JEE 1983
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 $P$ to $AB$ is ${x^{{2 \over 3}}} + {y^{{2 \over 3}}} = {c^{{2 \over 3}}}$
Correct Answer: Solve it.