A line which is passing through \[\left( 1,\text{ }2 \right)\] To Find: The equation of a straight line. By using the formula, The equation of line is \[[y\text{ }-\text{ }{{y}_{1}}~=\text{...

Given: A line which is passing through \[(2,\text{ }2\surd 3),\text{ }the\text{ }angle\text{ }is\text{ }{{75}^{o}}\] By using the formula, The equation of line is \[[y\text{ }-\text{...

Given: A straight line passing through the point \[\left( 0,\text{ }0 \right)\] and slope is \[m\] By using the formula, The equation of line is \[[y\text{ }-\text{ }{{y}_{1}}~=\text{ }m(x\text{...

Given: A line which is passing through \[\left(- 2,\text{ }3 \right),\]the angle is \[{{45}^{o}}\] By using the formula, The equation of line is \[[y\text{ }-\text{ }{{y}_{1}}~=\text{ }m(x\text{...

Given, A straight line passing through the point \[\left( 6,\text{ }2 \right)\]and the slope is \[-\text{ }3\] By using the formula, The equation of line is \[[y\text{ }-\text{ }{{y}_{1}}~=\text{...

Given: A line segment joining \[\left( 2,\text{ }-\text{ }5 \right)\text{ }and\text{ }\left( 1,\text{ }2 \right)\] if it cuts off an intercept \[-\text{ }4\] from y–axis By using the formula, The...

Given: The equation which makes an angle of \[tan{{~}^{\text{ }1}}\left( 3 \right)\] with the x–axis and cuts off an intercept of \[4\text{ }units\] on the negative direction of y–axis By using the...

There are two bisectors of the coordinate axes. Their inclinations with the positive x-axis are \[{{45}^{o}}~and\text{ }{{135}^{o}}\] The slope of the bisector is \[m\text{ }=\text{ }tan\text{...

With slope \[-\text{ }2\]and intersecting the x–axis at a distance of \[3\]units to the left of origin The slope is \[-\text{ }2\] and the coordinates are \[\left( -\text{ }3,\text{ }0 \right)\]...

(i) With slope \[2\] and y – intercept \[3\] The slope is \[2\]and the coordinates are \[\left( 0,\text{ }3 \right)\] Now, the required equation of line is \[y\text{ }=\text{ }mx\text{ }+\text{ }c\]...

Given: A line which makes an angle of \[{{150}^{o}}~\]with the x–axis and cutting off an intercept at \[2\] By using the formula, The equation of a line is \[y\text{ }=\text{ }mx\text{ }+\text{ }c\]...

Given: A line which is perpendicular and parallel to x–axis respectively and passing through \[\left( 4,\text{ }3 \right)\] By using the formula, The equation of line: \[[y\text{ }-\text{...

Given: \[x\text{ }=-\text{ }3,\text{ }x\text{ }=\text{ }2,\text{ }y\text{ }=-\text{ }2\] and \[y\text{ }=\text{ }3\] ∴ The Coordinates of the square are: \[A\left( 2,\text{ }3 \right),\text{...

Given: A line which is parallel to x–axis and having intercept \[-2\text{ }on\text{ }y\text{ }-\text{ }axis\] By using the formula, The equation of line: \[[y\text{ }-\text{ }{{y}_{1}}~=\text{...

Given: A line which is perpendicular to x–axis and having intercept \[-2\] By using the formula, The equation of line: \[[y\text{ }-\text{ }{{y}_{1}}~=\text{ }m(x\text{ }-\text{ }{{x}_{1}})]\] We...

Given: A line which is parallel to \[x-axis\] and passing through \[\left( 3,-\text{ }5 \right)\] By using the formula, The equation of line: \[[y\text{ }-\text{ }{{y}_{1}}~=\text{ }m(x\text{...

(i) \[A\text{ }\left( 4,\text{ }8 \right),\text{ }B\text{ }\left( 5,\text{ }12 \right),\text{ }C\text{ }\left( 9,\text{ }28 \right)\] By using the formula, The slope of the line \[=\text{...

(i) Which bisects the first quadrant angle? Given: Line bisects the first quadrant We know that, if the line bisects in the first quadrant, then the angle must be between line and the positive...

(i) Through \[\left( 5,\text{ }6 \right)\text{ }and\text{ }\left( 2,\text{ }3 \right)\] Through \[\left( 9,-\text{ }2 \right)\text{ }and\text{ }\left( 6,-\text{ }5 \right)\] By using the formula,...

(i) \[\left( -3,\text{ }2 \right)\text{ }and\text{ }\left( 1,\text{ }4 \right)\] By using the formula, ∴ The slope of the line is \[{\scriptscriptstyle 1\!/\!{ }_2}\]  ...

(i) \[-\text{ }\pi /4\] Let the slope of the line be \[m\] Where, \[m\text{ }=\text{ }tan\text{ }\theta \] So, the slope of Line is \[m\text{ }=\text{ }tan~\left( -\text{ }\pi /4 \right)~\]...

According to ques,: \[x\text{ }+\text{ }y\text{ }\text{ }4\text{ }=\text{ }0,\] \[3x\text{ }\text{ }7y\text{ }+\text{ }8\text{ }=\text{ }0,\] And \[4x\text{ }\text{ }y\text{ }\text{ }31\text{...

According to ques,: \[x\text{ }+\text{ }y\text{ }\text{ }4\text{ }=\text{ }0,\] \[3x\text{ }\text{ }7y\text{ }\text{ }8\text{ }=\text{ }0\] and \[4x\text{ }\text{ }y\text{ }\text{ }31\text{ }=\text{...

According to ques,: \[x\text{ }\text{ }5y\text{ }+\text{ }6\text{ }=\text{ }0,\] \[x\text{ }\text{ }3y\text{ }+\text{ }2\text{ }=\text{ }0\] and \[x\text{ }\text{ }2y\text{ }\text{ }3\text{ }=\text{...

to prove: Let us assume A (x1, y1) and B (x2, y2) be the given points and O be the origin. \[Slope\text{ }of\text{ }OA\text{ }=\text{ }{{m}_{1}}~=\text{ }{{y}_{1\times 1}}\] \[Slope\text{ }of\text{...

According to ques,:   Points (2, 0), (0, 3) and the line x + y = 1.   Let us assume A (2, 0), B (0, 3) be the given points.   Now, let us find the slopes \[Slope\text{ }of\text{...

To prove: The points: \[\left( 2,\text{ }-1 \right),\text{ }\left( 0,\text{ }2 \right),\text{ }\left( 2,\text{ }3 \right)\text{ }and\text{ }\left( 4,\text{ }0 \right)\] are the coordinates of the...

According to ques,:   The equations of the lines are   \[2x~-~y\text{ }+\text{ }3\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[x\text{ }+\text{ }y\text{ }+\text{ }2\text{...

(i) \[3x\text{ }+\text{ }y\text{ }+\text{ }12\text{ }=\text{ }0\text{ }and\text{ }x\text{ }+\text{ }2y\text{ }\text{ }1\text{ }=\text{ }0\]  According to ques,: The equations of the lines are...

According to ques,: The equation is perpendicular to: \[~\surd 3x\text{ }\text{ }y\text{ }+\text{ }5\text{ }=\text{ }0~\] equation and cuts off an intercept of 4 units with the negative direction of...

According to ques,: The vertices of ∆ABC are A (1, 4), B (− 3, 2) and C (− 5, − 3). Now let us find the slopes of ∆ABC. \[Slope\text{ }of\text{ }AB\text{ }=\text{ }\left[ \left( 2\text{ }\text{ }4...

According to ques,: A (1, 3) and B (3, 1) be the points joining the perpendicular bisector Let C be the midpoint of AB. hence, coordinates of C \[=\text{ }\left[ \left( 1+3 \right)/2,\text{ }\left(...

According to ques,: The equation of the line perpendicular to: \[~x~-~3y\text{ }+\text{ }5\text{ }=\text{ }0\] is \[3x\text{ }+\text{ }y\text{ }+~\lambda ~=\text{ }0,\] Where, λ is a constant. It...

Given: The equation of the line parallel to: \[3x~-~4y\text{ }+\text{ }5\text{ }=\text{ }0\] is \[3x\text{ }\text{ }4y\text{ }+~\lambda ~=\text{ }0,\] Where, λ is a constant. It passes through...

Given: \[{{L}_{1}}~=\text{ }\left( b\text{ }+\text{ }c \right)x\text{ }+\text{ }ay\text{ }+\text{ }1\text{ }=\text{ }0\] \[{{L}_{2}}~=\text{ }\left( c\text{ }+\text{ }a \right)x\text{ }+\text{...

Given: \[{{p}_{1}}x\text{ }+\text{ }{{q}_{1}}y\text{ }=\text{ }1\] \[{{p}_{2}}x\text{ }+\text{ }{{q}_{2}}y\text{ }=\text{ }1\] and , \[{{p}_{3}}x\text{ }+\text{ }{{q}_{3}}y\text{ }=\text{ }1\] The...

Given: \[{{m}_{1}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{1}}~=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[{{m}_{2}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{2}}~=\text{ }0\text{ }\ldots...

Given: \[2x~-~5y\text{ }+\text{ }3\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[5x~-~9y\text{ }+~\lambda ~=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)\] And , \[x~-~2y\text{...

\[\left( \mathbf{i} \right)~15x\text{ }\text{ }18y\text{ }+\text{ }1\text{ }=\text{ }0,\text{ }12x\text{ }+\text{ }10y\text{ }\text{ }3\text{ }=\text{ }0\] and \[6x\text{ }+\text{ }66y\text{ }\text{...

Given: \[y\text{ }=\text{ }\surd 3x\text{ }+\text{ }1\ldots \ldots \text{ }\left( 1 \right)\] \[y\text{ }=\text{ }4\text{ }\ldots \ldots .\text{ }\left( 2 \right)\] and \[y\text{ }=\text{ }\text{...

Given: \[3x\text{ }+\text{ }2y\text{ }+\text{ }6\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\] \[2x\text{ }-\text{ }5y\text{ }+\text{ }4\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 2...

Given: \[y\text{ }=\text{ }{{m}_{1}}x\text{ }+\text{ }{{c}_{1}}~\ldots \text{ }\left( 1 \right)\] \[y\text{ }=\text{ }{{m}_{2}}x\text{ }+\text{ }{{c}_{2}}~\ldots \text{ }\left( 2 \right)\] and ,...

\[\left( \mathbf{i} \right)~x\text{ }+\text{ }y\text{ }\text{ }4\text{ }=\text{ }0,\text{ }2x\text{ }\text{ }y\text{ }+\text{ }3\text{ }0\] and \[x\text{ }\text{ }3y\text{ }+\text{ }2\text{ }=\text{...

\[\left( \mathbf{i} \right)~2x\text{ }\text{ }y\text{ }+\text{ }3\text{ }=\text{ }0\] And \[x\text{ }+\text{ }y\text{ }\text{ }5\text{ }=\text{ }0\] Given: The equations of the lines are: \[2x\text{...

Given: The lines: \[4x\text{ }+\text{ }3y\text{ }+\text{ }10\text{ }=\text{ }0;\text{ }5x\text{ }\text{ }12y\text{ }+\text{ }26\text{ }=\text{ }0\] And \[7x\text{ }+\text{ }24y\text{ }=\text{ }50.\]...

Given: The normal forms of the lines: \[3x\text{ }-\text{ }4y\text{ }+\text{ }4\text{ }=\text{ }0\] And \[~2x\text{ }+\text{ }4y\text{ }-\text{ }5\text{ }=\text{ }0.\] To find, in given normal form...

Given: the equation is: \[~x/a\text{ }+\text{ }y/b\text{ }=\text{ }1~\] As , General equation of line \[~y\text{ }=\text{ }mx\text{ }+\text{ }c.\] \[bx\text{ }+\text{ }ay\text{ }=\text{ }ab\]...

\[\left( \mathbf{i} \right)~x\text{ }+\text{ }\surd 3y\text{ }\text{ }4\text{ }=\text{ }0\] \[x\text{ }+\text{ }\surd 3y\text{ }=\text{ }4\] The normal form of the given line, \[where\text{ }p\text{...

(iii) Given: \[\surd 3x\text{ }+\text{ }y\text{ }+\text{ }2\text{ }=\text{ }0~\] \[-\surd 3x\text{ }\text{ }y\text{ }=\text{ }2\]

(i) Given: \[\surd 3x\text{ }+\text{ }y\text{ }+\text{ }2\text{ }=\text{ }0~\] \[y\text{ }=\text{ }\text{ }\surd 3x\text{ }\text{ }2\] following is the slope intercept form of the given line....

The equation of the line that passes through P(x1, y1) and makes an angle of θ with the x–axis. To find the length of PQ. By using the formula, The equation of the line is given by:

Given: \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }=\text{ }A\text{ }\left( 4,\text{ }-1 \right)\] Let us find Coordinates of the two points on this line which are at a distance of 5 units...

Given: \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }=\text{ }A\text{ }\left( 2,\text{ }1 \right),\] \[\theta ~=~\pi /4\text{ }=\text{ }45{}^\circ \] Let us find the length AB. By using the...

Given: \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }=\text{ }A\text{ }\left( 3,\text{ }4 \right),\] \[~\theta ~=\text{ }\pi /6\text{ }=\text{ }30{}^\circ \] To find the length PQ. By using...

Given: \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\text{ }=\text{ }A\text{ }\left( 1,\text{ }2 \right),\text{ }\theta ~=\text{ }60{}^\circ \] To find the distance AP, On using the formula, The...

  Given: \[p\text{ }=\text{ }2,\text{ }sin\text{ }\alpha \text{ }=\text{ }1/3\] As  , \[~cos\text{ }\alpha \text{ }=\text{ }\surd \left( 1\text{ }\text{ }si{{n}^{2}}~\alpha  \right)\] \[=\text{...

Given: \[p\text{ }=\text{ }3,\text{ }\alpha \text{ }=\text{ }ta{{n}^{-1}}~\left( 5/12 \right)\] hence , \[tan\text{ }\alpha \text{ }=\text{ }5/12\] \[sin\text{ }\alpha \text{ }=\text{ }5/13\]...

Given: \[p\text{ }=\text{ }4,\text{ }\alpha \text{ }=\text{ }15{}^\circ \] The equation of the line in normal form is given by as, \[~cos\text{ }15{}^\circ ~=\text{ }cos\text{ }\left( 45{}^\circ...

Given: \[p\text{ }=\text{ }4,\text{ }\alpha \text{ }=\text{ }30{}^\circ \] The equation of the line in normal form is given by On using the formula, \[x\text{ }cos~\alpha \text{ }+\text{ }y\text{...

\[\left( \mathbf{i} \right)~p\text{ }=\text{ }5,\text{ }\alpha \text{ }=\text{ }60{}^\circ \] Given: \[p\text{ }=\text{ }5,\text{ }\alpha \text{ }=\text{ }60{}^\circ \] The equation of the line in...

Given: \[a\text{ }=\text{ }b\text{ }and\text{ }ab\text{ }=\text{ }25\] to find the equation of the line which cutoff intercepts on the axes. \[\therefore ~{{a}^{2}}~=\text{ }25\] \[a\text{ }=\text{...

Given: Intercepts cut off on the coordinate axes by the line \[~ax\text{ }+\text{ }by\text{ }+8\text{ }=\text{ }0\text{ }\ldots \ldots \text{ }\left( i \right)\] And are equal in length but opposite...

(i) Equal in magnitude and both positive Given: \[a\text{ }=\text{ }b\] to find the equation of line cutoff intercepts from the axes. On using the formula, The equation of the line is: \[x/a\text{...

Given: A line passing through (1, -2) Let the equation of the line cutting equal intercepts at coordinates of length ‘a’ is By using the formula, The equation of the line is: \[~x/a\text{ }+\text{...

 (i) Cutting off intercepts 3 and 2 from the axes. Given: \[a\text{ }=\text{ }3,\] \[~b\text{ }=\text{ }2\] to find the equation of line cutoff intercepts from the axes. on using the formula, The...

Given: The vertices of triangle ABC are: \[A\text{ }\left( -1,\text{ }-2 \right),\text{ }B\left( 0,\text{ }1 \right)\text{ }and\text{ }C\left( 2,\text{ }0 \right).\] Let us find the equation of...

Given: The rectangle formed by the lines, \[x\text{ }=\text{ }a,\text{ }x\text{ }=\text{ }a,\text{ }y\text{ }=\text{ }b\text{ }and\text{ }y\text{ }=\text{ }b\] It is clear that, the vertices of the...

\[A\text{ }\left( -1,\text{ }6 \right),\text{ }B\text{ }\left( -3,\text{ }-9 \right)\text{ }and\text{ }C\text{ }\left( 5,\text{ }-8 \right)\] be the coordinates of the given triangle. Let: D, E, and...

\[~\left( \mathbf{i} \right)~\left( 1,\text{ }4 \right),\text{ }\left( 2,\text{ }-3 \right)\text{ }and\text{ }\left( -1,\text{ }-2 \right)\] Given: Points A (1, 4), B (2, -3) and C (-1, -2). Let ,...

The equation of the straight line passing through the points of intersection of 2x + 3y + 1 = 0 and 3x − 5y − 5 = 0 is \[\begin{array}{*{35}{l}} 2x\text{ }+\text{ }3y\text{ }+\text{ }1\text{...

\[x\text{ }+\text{ }2y\text{ }+\text{ }3\text{ }=\text{ }0\text{ }and\text{ }3x\text{ }+\text{ }4y\text{ }+\text{ }7\text{ }=\text{ }0\] The equation of the straight line passing through the points...

The equation passes through (2,-1) and make an angle of 45° with the line 6x + 5y – 8 = 0 Since, the equations of two lines passing through a point x1, y1 and making an angle α with the given line y...

The given lines are \[\begin{array}{*{35}{l}} 3x~-~4y\text{ }+\text{ }a\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)  \\ 3x~-~4y\text{ }+\text{ }3a\text{ }=\text{ }0\text{ }\ldots \text{...

9x + 6y – 7 = 0 and 3x + 2y + 6 = 0 are parallel lines The given equations of the lines can be written as: \[\begin{array}{*{35}{l}} 3x\text{ }+\text{ }2y\text{ }-\text{ }7/3\text{ }=\text{...

The lines 24x + 7y = 20 and 4x – 3y – 2 = 0 Let us assume, P(a, b) be any point on 2x + 11y − 5 = 0 So, 2a + 11b − 5 = 0