Straight Lines and Pair of Straight Lines
87 Questions
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.
1983
JEE Advanced
Numerical
IIT-JEE 1983
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 $$ABC$ is $\left| {{{x + y - 2} \over 7}} \right|$
Correct Answer: Solve it.
1983
JEE Advanced
Numerical
IIT-JEE 1983
Given the points $A\left( {0,4} \right)$ and $B\left( {0, - 4} \right)$, the equation of the locus of the point $P\left( {x,y} \right)$ such that $\left| {AP - BP} \right| = 6$ is .............
Correct Answer: $${{{y^2}} \over 9} - {{{x^2}} \over 7} = 1$$
1983
JEE Advanced
MCQ
IIT-JEE 1983
The straight line $5x + 4y = 0$ passes through the point of intersection of the straight lines $x + 2y - 10 = 0$ and $2x + y + 5 = 0.$
A.
TRUE
B.
FALSE
1982
JEE Advanced
Numerical
IIT-JEE 1982
$y = {10^x}$ is the reflection of ${\log _{10}}\,x$ in the line whose equation is ...........
Correct Answer: $$y = x$$
1982
JEE Advanced
Numerical
IIT-JEE 1982
The set of lines $ax + by + c = 0,$ where $3a + 2b + 4c = 0$ is concurrent at the point ..........
Correct Answer: $$\left( {{3 \over 4},{1 \over 2}} \right)$$
1981
JEE Advanced
Numerical
IIT-JEE 1981
The area enclosed within the curve $\left| x \right| + \left| y \right| = 1$ is .................
Correct Answer: 2 sq. units
1980
JEE Advanced
MCQ
IIT-JEE 1980
The point $\,\left( {4,\,1} \right)$ undergoes the following three transformations successively.
Reflection about the line $y=x$.
Translation through a distance 2 units along the positive direction of x-axis.
Rotation through an angle $p/4$ about the origin in the counter clockwise direction.
Then the final position of the point is given by the coordinates.
Reflection about the line $y=x$.
Translation through a distance 2 units along the positive direction of x-axis.
Rotation through an angle $p/4$ about the origin in the counter clockwise direction.
Then the final position of the point is given by the coordinates.
A.
$\left( {{1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right)$
B.
$\left( { - \sqrt 2 ,\,7\sqrt 2 } \right)$
C.
$\left( { - {1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right)$
D.
$\left( { \sqrt 2 ,\,7\sqrt 2 } \right)$
1980
JEE Advanced
Numerical
IIT-JEE 1980
A straight line $L$ is perpendicular to the line $5x - y = 1.$ The area of the triangle formed by the line $L$ and the coordinate axes is $5$. Find the equation of the Line $L$.
Correct Answer: $$x + 5y - 5\sqrt 2 = 0$$ or $$x + 5y + 5\sqrt 2 = 0$$
1979
JEE Advanced
MCQ
IIT-JEE 1979
The points $\left( { - a,\, - b} \right),\,\left( {0,\,0} \right),\,\left( {a,\,b} \right)$ and $\left( {{a^2},\,ab} \right)$ are :
A.
Collinear
B.
Vertices of a parallelogram
C.
Vertices of a rectangle
D.
None of these
1979
JEE Advanced
Numerical
IIT-JEE 1979
(a) Two vertices of a triangle are $(5, -1)$ and $(-2, 3).$ If the orthocentre of the triangle is the origin, find the coordinates of the third point.
(b) Find the equation of the line which bisects the obtuse angle between the lines $x - 2y + 4 = 0$ and $4x - 3y + 2 = 0$.
(b) Find the equation of the line which bisects the obtuse angle between the lines $x - 2y + 4 = 0$ and $4x - 3y + 2 = 0$.
Correct Answer: (a) $$(-4, -7)$$
<br>(b) $$\left( {4 - \sqrt 5 } \right)x + \left( {2\sqrt 5 - 3} \right)y - \left( {4\sqrt 5 - 2} \right) = 0$$
1978
JEE Advanced
Numerical
IIT-JEE 1978
A straight line segment of length $\ell $ moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio $1 : 2$
Correct Answer: $$9{x^2} + 36{y^2} = 4{\ell ^2}$$
1978
JEE Advanced
Numerical
IIT-JEE 1978
One side of rectangle lies along the line $4x + 7y + 5 = 0.$ Two of its vertices are $(-3, 1)$ and $(1, 1).$ Find the equations of the other three sides.
Correct Answer: $$\matrix{
{4x + 7y - 11 = 0} \cr
{7x - 4y - 3 = 0} \cr
{7x - 4y + 25 = 0} \cr
} $$
1978
JEE Advanced
Numerical
IIT-JEE 1978
The area of a triangle is $5$. Two of its vertices are $A\left( {2,1} \right)$ and $B\left( {3, - 2} \right)$. The third vertex $C$ lies on $y = x + 3$. Find $C$.
Correct Answer: $$\left( {{{ - 3} \over 2},{3 \over 2}} \right)$$ 0r $$\left( {{{ 7} \over 2},{13 \over 2}} \right)$$