\displaystyle \textbf{Co-ordinate Geometry}

\displaystyle \textbf{Introduction}
\displaystyle \text{Co-ordinate Geometry is the branch of mathematics in which a pair of two numbers,}
\displaystyle \text{called co-ordinates, is used to represent the position of a point with respect to}
\displaystyle \text{two mutually perpendicular number lines called co-ordinate axes.}
\displaystyle \text{The location of points comes under the heading co-ordinate and their relations,}
\displaystyle \text{with respect to different figures, come under the heading geometry.}
\displaystyle \text{Together, the location of the points and their relationship with different}
\displaystyle \text{geometrical figures is called Co-ordinate Geometry.}

\displaystyle \textbf{Dependent and Independent Variables}
\displaystyle \text{In linear equations such as }3x+4y=5,\quad x-3y+8=0,\quad y=mx+c,
\displaystyle x=5y-8,\text{ etc., the letters }x\text{ and }y\text{ are called variables.}
\displaystyle \textbf{1. }\text{If a linear equation in }x\text{ and }y\text{ is expressed with }y
\displaystyle \text{as the subject of the equation, }y\text{ is called the dependent variable}
\displaystyle \text{and }x\text{ is called the independent variable.}
\displaystyle \text{Examples:}
\displaystyle \text{(i) }y=3x-6\qquad\text{(ii) }y=5-\frac{x}{4}
\displaystyle \text{(iii) }y=2(3x-5)+7
\displaystyle \text{In each of these equations, }y\text{ is dependent and }x\text{ is independent.}
\displaystyle \textbf{2. }\text{If a linear equation in }x\text{ and }y\text{ is expressed with }x
\displaystyle \text{as the subject of the equation, }x\text{ is called the dependent variable}
\displaystyle \text{and }y\text{ is called the independent variable.}
\displaystyle \text{Examples:}
\displaystyle \text{(i) }x=5y+7\qquad\text{(ii) }x=5(5y+8)-10
\displaystyle \text{(iii) }x=7-\frac{2y}{3}
\displaystyle \text{In each of these equations, }x\text{ is dependent and }y\text{ is independent.}

\displaystyle \textbf{Ordered Pair}
\displaystyle \text{An ordered pair means a pair of two objects taken in a specific order.}
\displaystyle \text{In relation to co-ordinate geometry, an ordered pair means a pair of two numbers}
\displaystyle \text{in which the order is important and necessary.}
\displaystyle \textbf{1. }\text{To form an ordered pair, the numbers are written in a specific order,}
\displaystyle \text{separated by a comma and enclosed in small brackets.}
\displaystyle \text{Examples of ordered pairs are:}
\displaystyle (5,7),\quad(-6,8),\quad(0,0),\quad(0,-6),\quad(5,0),\quad\left(3\frac{1}{2},-2\right).
\displaystyle \textbf{2. }\text{In the ordered pair }(a,b),\ a\text{ is called its first component}
\displaystyle \text{and }b\text{ is called its second component.}
\displaystyle \textbf{3. }\text{Ordered pairs }(5,7)\text{ and }(7,5)\text{ are different, i.e.}
\displaystyle (5,7)\ne(7,5).
\displaystyle \textbf{4. }\text{If two ordered pairs are equal, their corresponding components are equal.}
\displaystyle (a,b)=(c,d)\Rightarrow a=c\text{ and }b=d.
\displaystyle \textbf{5. }\text{An ordered pair can have both of its components equal.}
\displaystyle \text{For example, }(5,5),\quad(-6,-6),\quad(0,0),\text{ etc.}

\displaystyle \textbf{Cartesian Plane}
\displaystyle \text{A Cartesian (or a co-ordinate) plane consists of two mutually perpendicular}
\displaystyle \text{number lines intersecting each other at their zeros.}
\displaystyle \text{The two mutually perpendicular number lines }XOX'\text{ and }YOY'
\displaystyle \text{intersect each other at their zero }O.
\displaystyle \textbf{1. }\text{The horizontal number line }XOX'\text{ is called the }x\text{-axis.}
\displaystyle \textbf{2. }\text{The vertical number line }YOY'\text{ is called the }y\text{-axis.}
\displaystyle \textbf{3. }\text{The point of intersection }O\text{ is called the origin,}
\displaystyle \text{which is zero for both the axes.}
\displaystyle \text{The system consisting of the }x\text{-axis, the }y\text{-axis and the origin}
\displaystyle \text{is called the Cartesian co-ordinate system.}
\displaystyle \text{The }x\text{-axis and the }y\text{-axis together are called co-ordinate axes.}

\displaystyle \textbf{Co-ordinates of Points}
\displaystyle \text{The position of each point in a co-ordinate plane is determined by means}
\displaystyle \text{of an ordered pair with reference to the co-ordinate axes.}
\displaystyle \textbf{(i) }\text{Starting from the origin }O,\text{ measure the distance of the point}
\displaystyle \text{along the }x\text{-axis. This is called the }x\text{-coordinate or abscissa.}
\displaystyle \textbf{(ii) }\text{Starting from the origin }O,\text{ measure the distance of the point}
\displaystyle \text{along the }y\text{-axis. This is called the }y\text{-coordinate or ordinate.}
\displaystyle \text{Thus, co-ordinates of a point}
\displaystyle =\text{ position of the point with reference to the co-ordinate axes}
\displaystyle =(\text{abscissa},\text{ ordinate}).
\displaystyle \text{In stating the co-ordinates of a point, the abscissa precedes the ordinate,}
\displaystyle \text{and both are enclosed in small brackets and separated by a comma.}
\displaystyle \text{If the abscissa is }x\text{ and the ordinate is }y,\text{ the co-ordinates are }(x,y).

\displaystyle \textbf{Quadrants and Sign Convention} \displaystyle \textbf{1. Quadrants}
\displaystyle \text{The co-ordinate axes divide a co-ordinate plane into four parts,}
\displaystyle \text{which are known as quadrants.}
\displaystyle \text{Each point in the plane lies either in one of the quadrants or on one of the axes.}
\displaystyle \text{Starting from }OX\text{ in the anti-clockwise direction:}
\displaystyle XOY\text{ is called the first quadrant.}
\displaystyle YOX'\text{ is called the second quadrant.}
\displaystyle X'OY'\text{ is called the third quadrant.}
\displaystyle Y'OX\text{ is called the fourth quadrant.}
\displaystyle \textbf{2. Sign Convention}
\displaystyle \textbf{(i) First quadrant: }\text{abscissa is positive and ordinate is positive }(+,+).
\displaystyle \textbf{(ii) Second quadrant: }\text{abscissa is negative and ordinate is positive }(-,+).
\displaystyle \textbf{(iii) Third quadrant: }\text{abscissa is negative and ordinate is negative }(-,-).
\displaystyle \textbf{(iv) Fourth quadrant: }\text{abscissa is positive and ordinate is negative }(+,-).

\displaystyle \textbf{Plotting of Points} \displaystyle \text{The position of a point in a co-ordinate plane can be represented by plotting}
\displaystyle \text{its ordered pair }(x,y)\text{ with reference to the co-ordinate axes.}
\displaystyle \text{On a graph paper, draw the co-ordinate axes }XOX'\text{ and }YOY'
\displaystyle \text{intersecting at the origin }O\text{ and mark a suitable scale on both axes.}
\displaystyle \text{For plotting any point, two steps are followed.}
\displaystyle \text{For example, to plot the point }A(4,2):
\displaystyle \textbf{Step 1: }\text{Starting from the origin }O,\text{ move }4\text{ units along}
\displaystyle \text{the positive direction of the }x\text{-axis, i.e. to the right of }O.
\displaystyle \textbf{Step 2: }\text{From there, move }2\text{ units upwards, parallel to the positive}
\displaystyle \text{direction of the }y\text{-axis, and mark the point reached as }A(4,2).
\displaystyle \text{Similarly, points }B(-5,3),\ C(-4,-5)\text{ and }D(5,-2)\text{ can be plotted.}
\displaystyle \textbf{Important Points:}
\displaystyle \textbf{1. }\text{The co-ordinates of the origin are }(0,0).
\displaystyle \textbf{2. }\text{For a point on the }x\text{-axis, its ordinate is always zero.}
\displaystyle \text{Hence, the co-ordinates of a point on the }x\text{-axis are of the form }(x,0).
\displaystyle \text{Examples: }(7,0),\ (3,0),\ (0,0),\ (-4,0),\ (-8,0).
\displaystyle \textbf{3. }\text{For a point on the }y\text{-axis, its abscissa is always zero.}
\displaystyle \text{Hence, the co-ordinates of a point on the }y\text{-axis are of the form }(0,y).
\displaystyle \text{Examples: }(0,8),\ (0,3),\ (0,0),\ (0,-2),\ (0,-5).

\displaystyle \textbf{Graphs of }x=0,\ y=0,\ x=a,\ y=a,\textbf{ etc.} \displaystyle \textbf{1. }\ x=0\text{ is the equation of the }y\text{-axis, since the value of }x
\displaystyle \text{for every point }(x,y)\text{ on the }y\text{-axis is zero.}
\displaystyle \text{For example, }(0,7),\ (0,0),\ (0,-8),\ (0,15)
\displaystyle \text{all lie on the }y\text{-axis since their abscissa is }x=0.
\displaystyle \textbf{2. }\ x=a\text{ is the equation of a line parallel to the }y\text{-axis}
\displaystyle \text{and at a distance of }|a|\text{ units from the }y\text{-axis.}
\displaystyle \text{For example, the equation of the line }AB\text{ is }x=-5,
\displaystyle \text{i.e. }x+5=0.
\displaystyle \text{The equation of the line }CD\text{ is }x=-2,\text{ i.e. }x+2=0.
\displaystyle \text{The equation of the line }EF\text{ is }x=3.
\displaystyle \text{The equation of the line }GH\text{ is }x=6.
\displaystyle \textbf{3. }\ y=0\text{ is the equation of the }x\text{-axis, since the value of }y
\displaystyle \text{for every point }(x,y)\text{ on the }x\text{-axis is zero.}
\displaystyle \textbf{4. }\ y=a\text{ is the equation of a line parallel to the }x\text{-axis}
\displaystyle \text{and at a distance of }|a|\text{ units from the }x\text{-axis.}

\displaystyle \textbf{Graphing a Linear Equation}
\displaystyle \text{If the graph of an equation is a straight line, the equation is called}
\displaystyle \text{a linear equation.}
\displaystyle \text{To draw the graph of a linear equation:}
\displaystyle \textbf{(i) }\text{Plot a few points which satisfy the given equation.}
\displaystyle \textbf{(ii) }\text{Draw a straight line passing through these points.}

\displaystyle \textbf{Inclination and Slope} \displaystyle \textbf{1. Inclination}
\displaystyle \text{The angle which a straight line makes with the positive direction of the }x\text{-axis,}
\displaystyle \text{measured in the anti-clockwise direction, is called the inclination of the line.}
\displaystyle \text{The inclination of a line is usually denoted by }\theta.
\displaystyle \text{For example, the inclination of a line may be }\theta=45^\circ,\ 130^\circ,\text{ etc.}
\displaystyle \textbf{1. }\text{For the }x\text{-axis and every line parallel to the }x\text{-axis,}
\displaystyle \text{the inclination is zero, i.e. }\theta=0^\circ.
\displaystyle \textbf{2. }\text{For the }y\text{-axis and every line parallel to the }y\text{-axis,}
\displaystyle \text{the inclination is }90^\circ,\text{ i.e. }\theta=90^\circ.
\displaystyle \textbf{2. Slope (Gradient)}
\displaystyle \text{If }\theta\text{ is the inclination of a line, its slope is }\tan\theta
\displaystyle \text{and is usually denoted by }m.
\displaystyle \therefore\ \text{Slope }=m=\tan\theta.
\displaystyle \text{If the inclination of a line is }30^\circ,\text{ then}
\displaystyle m=\tan30^\circ=\frac{1}{\sqrt{3}}.
\displaystyle \text{If the inclination of a line is }45^\circ,\text{ then}
\displaystyle m=\tan45^\circ=1.
\displaystyle \textbf{1. }\text{For the }x\text{-axis and every line parallel to the }x\text{-axis,}
\displaystyle \theta=0^\circ\Rightarrow m=\tan0^\circ=0.
\displaystyle \textbf{2. }\text{For the }y\text{-axis and every line parallel to the }y\text{-axis,}
\displaystyle \theta=90^\circ\Rightarrow m=\tan90^\circ=\text{not defined.}

\displaystyle \textbf{Y-Intercept} \displaystyle \text{If a straight line meets the }y\text{-axis at a point, the directed distance}
\displaystyle \text{of this point from the origin is called the }y\text{-intercept.}
\displaystyle \text{The }y\text{-intercept is usually denoted by }c.
\displaystyle \textbf{1. }\text{For the }x\text{-axis, the }y\text{-intercept is }0.
\displaystyle \textbf{2. }\text{For a line parallel to the }y\text{-axis, a }y\text{-intercept exists only}
\displaystyle \text{when the line is the }y\text{-axis itself; otherwise it does not meet the }y\text{-axis.}
\displaystyle \textbf{3. }\text{The }y\text{-intercept is positive if the line meets the }y\text{-axis}
\displaystyle \text{above the origin, and negative if it meets the }y\text{-axis below the origin.}
\displaystyle \text{For example, if the line meets the }y\text{-axis at }y=5,\text{ then }c=5.
\displaystyle \text{If the line meets the }y\text{-axis at }y=-8,\text{ then }c=-8.

\displaystyle \textbf{Finding the Slope and the Y-Intercept of a Given Line}
\displaystyle \textbf{Steps:}
\displaystyle \textbf{1. }\text{Let the equation of the given line be }ax+by+c=0.
\displaystyle \textbf{2. }\text{Make }y\text{ the subject of the equation.}
\displaystyle ax+by+c=0
\displaystyle \Rightarrow by=-ax-c
\displaystyle \Rightarrow y=-\frac{a}{b}x-\frac{c}{b}
\displaystyle \textbf{3. }\text{Comparing with the slope-intercept form }y=mx+c,
\displaystyle \text{the coefficient of }x\text{ gives the slope and the constant term}
\displaystyle \text{gives the }y\text{-intercept.}
\displaystyle \therefore\ \text{Slope }(m)=-\frac{a}{b}
\displaystyle \text{and }y\text{-intercept}=-\frac{c}{b}.

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