Various Forms of Equations of a Line

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JEE Main 2016

Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7 x - y - 5 = 0$ . If its diagonals intersect at $(1, 2)$ , then which one of the following is a vertex of this rhombus?

(\frac{1}{3}, \frac{8}{3})

(B)

(\frac{10}{3}, \frac{7}{3})

(C)

(- 3, - 9)

(D)

(- 3, - 8)

JEE Main 2015

LEVELJEE Main

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0), (0, 41)$ and $(41, 0)$ is

Let $P S$ 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 $P S$ is

(A)

4 x + 7 y + 3 = 0

(B)

2 x - 9 y - 11 = 0

(C)

4 x - 7 y - 11 = 0

(D)

2 x + 9 y + 7 = 0

JEE Main 2014

LEVELJEE Main

Let $a, b, c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4 a x + 2 a y + c = 0$ and $5 b x + 2 b y + d = 0$ lies in the fourth quadrant and is equidistant from the two axes then

(A)

3 b c - 2 a d = 0

(B)

3 b c + 2 a d = 0

(C)

2 b c - 3 a d = 0

(D)

2 b c + 3 a d = 0

JEE Main 2013

LEVELJEE Main

A ray of light along $x + 3 y = 3$ gets reflected upon reaching x-axis, the equation of the reflected ray is

(A)

y = x + 3

(B)

3 y = x - 3

(C)

y = 3 x - 3

(D)

3 y = x - 1

JEE Main 2012

LEVELJEE Main

A line is drawn through the point $(1, 2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $O P Q$ , where $O$ is the origin. If the area of the triangle $O P Q$ is least, then the slope of the line $P Q$ is

The perpendicular bisector of the line segment joining $P (1, 4)$ and $Q (k, 3)$ has y-intercept $- 4$ . Then a possible value of $k$ is

Let $P = (- 1, 0), Q = (0, 0)$ and $R = (3, 33)$ be three points. The equation of the bisector of the angle $P QR$ is

(A)

\frac{3}{2} x + y = 0

(B)

x + 3 y = 0

(C)

3 x + y = 0

(D)

x + \frac{3}{2} y = 0

JEE Main 2006

LEVELJEE Main

A straight line through the point $A (3, 4)$ is such that its intercept between the axes is bisected at $A$ . Its equation is

(A)

x + y = 7

(B)

3 x - 4 y + 7 = 0

(C)

4 x + 3 y = 24

(D)

3 x + 4 y = 25

JEE Main 2006

LEVELJEE Main

If $(a, a^{2})$ falls inside the angle made by the lines $y = \frac{x}{2}$ , $x > 0$ and $y = 3 x$ , $x > 0$ , then $a$ belong to

(A)

(0, \frac{1}{2})

(B)

(3, \infty)

(C)

(\frac{1}{2}, 3)

(D)

(- 3, - \frac{1}{2})

JEE Main 2004

LEVELJEE Main

The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the co-ordinate axes whose sum is $- 1$ is

(A)

\frac{x}{2} - \frac{y}{3} = 1

and

\frac{x}{- 2} + \frac{y}{1} = 1

(B)

\frac{x}{2} - \frac{y}{3} = - 1

and

\frac{x}{- 2} + \frac{y}{1} = - 1

(C)

\frac{x}{2} + \frac{y}{3} = 1

and

\frac{x}{- 2} + \frac{y}{- 1} = 1

(D)

\frac{x}{2} + \frac{y}{- 3} = - 1

and

\frac{x}{- 2} + \frac{y}{1} = - 1

JEE Main 2004

LEVELBoard

If the sum of the slopes of the lines given by $x^{2} - 2 c x y - 7 y^{2} = 0$ is four times their product $c$ has the value

If one of the lines given by $6 x^{2} - x y + 4 c y^{2} = 0$ is $3 x + 4 y = 0$ , then $c$ equals

A square of side $a$ lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an angle $α$ ( $0 < α < \frac{π}{4}$ ) with the positive direction of x-axis. The equation of its diagonal not passing through the origin is

(A)

y (cos α + sin α) + x (cos α - sin α) = a

(B)

y (cos α - sin α) - x (sin α - cos α) = a

(C)

y (cos α + sin α) + x (sin α - cos α) = a

(D)

y (cos α + sin α) + x (sin α + cos α) = a

JEE Main 2002

LEVELJEE Main

Locus of mid point of the portion between the axes of $x cos α + y sin α = p$ where $p$ is constant is

(A)

x^{2} + y^{2} = \frac{4}{p ^{2}}

(B)

x^{2} + y^{2} = 4 p^{2}

(C)

\frac{1}{x ^{2}} + \frac{1}{y ^{2}} = \frac{2}{p ^{2}}

(D)

\frac{1}{x ^{2}} + \frac{1}{y ^{2}} = \frac{4}{p ^{2}}

JEE Main 2002

LEVELJEE Main

If the pair of lines $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ intersect on the y-axis then

(A)

2 f g h = b g^{2} + c h^{2}

(B)

b g^{2} \neq = c h^{2}

(C)

ab c = 2 f g h

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

Let $P = (- 1, 0), Q = (0, 0)$ and $R = (3, 33)$ be three points. Then the equation of the bisector of the angle $P QR$ is

(A)

\frac{3}{2} x + y = 0

(B)

x + 3 y = 0

(C)

3 x + y = 0

(D)

x + \frac{3}{2} y = 0

JEE Advanced 2001

LEVELJEE Main

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

Area of the parallelogram formed by the lines $y = m x, y = m x + 1, y = n x$ and $y = n x + 1$ equals

(A)

∣ m + n ∣/ (m - n)^{2}

(B)

2/∣ m + n ∣

(C)

1/∣ m + n ∣

(D)

1/∣ m - n ∣

JEE Advanced 1994

LEVELJEE Main

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

(A)

x + 4 y = 13, y = 4 x - 7

(B)

4 x + y = 13, 4 y = x - 7

(C)

4 x + y = 13, y = 4 x - 7

(D)

y - 4 x = 13, y + 4 x = 7

JEE Advanced 1988

LEVELBoard

If $P = (1, 0), Q = (- 1, 0)$ and $R = (2, 0)$ are three given points, then locus of the point $S$ satisfying the relation $S Q^{2} + S R^{2} = 2 S P^{2}$ , 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.

JEE Advanced 1986

LEVELJEE Main

All points lying inside the triangle formed by the points $(1, 3), (5, 0)$ and $(- 1, 2)$ satisfy

* Multiple Correct Options

(A)

3 x + 2 y \geq 0

(B)

2 x + y - 13 \geq 0

(C)

2 x - 3 y - 12 \leq 0

(D)

- 2 x + y \geq 0

The straight lines $x + y = 0, 3 x + y - 4 = 0, x + 3 y - 4 = 0$ form a triangle which is

The centre of circle inscribed in square formed by the lines $x^{2} - 8 x + 12 = 0$ and $y^{2} - 14 y + 45 = 0$ , is

(A)

(4, 7)

(B)

(7, 4)

(C)

(9, 4)

(D)

(4, 9)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two sides of a rhombus are along the lines, x−y+1=0 and 7x−y−5=0. If its diagonals intersect at (1,2), then which one of the following is a vertex of this rhombus?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0,0),(0,41) and (41,0) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A ray of light along x+3​y=3​ gets reflected upon reaching x-axis, the equation of the reflected ray is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The perpendicular bisector of the line segment joining P(1,4) and Q(k,3) has y-intercept −4. Then a possible value of k is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P=(−1,0),Q=(0,0) and R=(3,33​) be three points. The equation of the bisector of the angle PQR is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A straight line through the point A(3,4) is such that its intercept between the axes is bisected at A. Its equation is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (a,a2) falls inside the angle made by the lines y=2x​, x>0 and y=3x, x>0, then a belong to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the straight line passing through the point (4,3) and making intercepts on the co-ordinate axes whose sum is −1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the sum of the slopes of the lines given by x2−2cxy−7y2=0 is four times their product c has the value

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one of the lines given by 6x2−xy+4cy2=0 is 3x+4y=0, then c equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Locus of mid point of the portion between the axes of xcosα+ysinα=p where p is constant is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the pair of lines ax2+2hxy+by2+2gx+2fy+c=0 intersect on the y-axis then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0,0),(0,21) and (21,0), is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P=(−1,0),Q=(0,0) and R=(3,33​) be three points. Then the equation of the bisector of the angle PQR is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Area of the parallelogram formed by the lines y=mx,y=mx+1,y=nx and y=nx+1 equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equations to a pair of opposite sides of parallelogram are x2−5x+6=0 and y2−6y+5=0, the equations to its diagonals are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P=(1,0),Q=(−1,0) and R=(2,0) are three given points, then locus of the point S satisfying the relation SQ2+SR2=2SP2, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } All points lying inside the triangle formed by the points (1,3),(5,0) and (−1,2) satisfy

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The straight lines x+y=0,3x+y−4=0,x+3y−4=0 form a triangle which is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The centre of circle inscribed in square formed by the lines x2−8x+12=0 and y2−14y+45=0, is

Two sides of a rhombus are along the lines, $x - y + 1 = 0$ and $7 x - y - 5 = 0$ . If its diagonals intersect at $(1, 2)$ , then which one of the following is a vertex of this rhombus?

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0), (0, 41)$ and $(41, 0)$ is

Let $P S$ 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 $P S$ is

A ray of light along $x + 3 y = 3$ gets reflected upon reaching x-axis, the equation of the reflected ray is

The perpendicular bisector of the line segment joining $P (1, 4)$ and $Q (k, 3)$ has y-intercept $- 4$ . Then a possible value of $k$ is

Let $P = (- 1, 0), Q = (0, 0)$ and $R = (3, 33)$ be three points. The equation of the bisector of the angle $P QR$ is

A straight line through the point $A (3, 4)$ is such that its intercept between the axes is bisected at $A$ . Its equation is

If $(a, a^{2})$ falls inside the angle made by the lines $y = \frac{x}{2}$ , $x > 0$ and $y = 3 x$ , $x > 0$ , then $a$ belong to

The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the co-ordinate axes whose sum is $- 1$ is

If the sum of the slopes of the lines given by $x^{2} - 2 c x y - 7 y^{2} = 0$ is four times their product $c$ has the value

If one of the lines given by $6 x^{2} - x y + 4 c y^{2} = 0$ is $3 x + 4 y = 0$ , then $c$ equals

Locus of mid point of the portion between the axes of $x cos α + y sin α = p$ where $p$ is constant is

If the pair of lines $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ intersect on the y-axis then

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

Let $P = (- 1, 0), Q = (0, 0)$ and $R = (3, 33)$ be three points. Then the equation of the bisector of the angle $P QR$ is

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

Area of the parallelogram formed by the lines $y = m x, y = m x + 1, y = n x$ and $y = n x + 1$ equals

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

If $P = (1, 0), Q = (- 1, 0)$ and $R = (2, 0)$ are three given points, then locus of the point $S$ satisfying the relation $S Q^{2} + S R^{2} = 2 S P^{2}$ , is

All points lying inside the triangle formed by the points $(1, 3), (5, 0)$ and $(- 1, 2)$ satisfy

The straight lines $x + y = 0, 3 x + y - 4 = 0, x + 3 y - 4 = 0$ form a triangle which is

The centre of circle inscribed in square formed by the lines $x^{2} - 8 x + 12 = 0$ and $y^{2} - 14 y + 45 = 0$ , is