Co-ordinate Geometry

Coordinate geometry is used to represent algebraic relations on graphs.
We shall be dealing with two-dimensional problems, where there are two variables to be handled.  The variables are normally denoted by the ordered pair (x, y).
The horizontal axis is the X -axis and the vertical axis is the Y-axis. If the coordinates of a point on the XY plane is (x, y), it implies that it is at a perpendicular distance of x from the Y-axis and at a perpendicular distance y from the X-axis. The point of intersection of the X and Y-axis is called the origin and the coordinates of this point is (0, 0). Signs of the coordinate ‘x’ and ‘y’ depends on the quadrant in which the point lies

I quadrant :        x coordinate is positive; y coordinate is positive
II quadrant : x coordinate is negative; y coordinate is positive
III quadrant :     x coordinate is negative: y coordinate is negative
IV quadrant : x coordinate is positive; y coordinate is negative

Some fundamental formulae: 

1. Distance between the points (x1,y1)$\left(x_{1,}{y}_1\right)$ and (x2,y2)$\left(x_{2,}{y}_2\right)$ is =(x2−x1)2+(y2−y1)2−−−−−−−−−−−−−−−−−−√$=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Slope: 
Angle made by the line with the positive direction of x - axis is called the inclination of the line.
If θ$\theta$ is the inclination, then ‘tan θ$\theta$’ denotes the slope of the line.
Slope of the line joining the points (x1,y1)$\left(x_{1,}{y}_1\right)$ and (x2,y2)$\left(x_{2,}{y}_2\right)$ is y2−y1x2−x1${\displaystyle \frac{y_2-y_1}{x_2-x_1}}$; . The slope is also indicated by m.

Slope intercept form: 
All straight lines can be written as y = mx + c,  where m is the slope of the straight line, c is the Y intercept or the Y coordinate of the point at which the straight line cuts the Y-axis.

Point slope form:
The equation of a straight line passing through (x1,y1)$\left(x_{1,}{y}_1\right)$ and having a slope m is y−y1=m(x−x1)$y-y_1=m\left(x-x_1\right)$

Two point form:
The equation of a straight line passing through two points (x1,y1)$\left(x_{1,}{y}_1\right)$ and (x2,y2)$\left(x_{2,}{y}_2\right)$ is y−y1=y2−y1x2−x1(x−x1)$y-y_1={\displaystyle \frac{y_2-y_1}{x_2-x_1}(x-x_1)}$

Intercept form:
The intercept form of a line is xa+yb=1${\displaystyle \frac{x}{a}+\frac{y}{b}=1}$
Where ‘a’ is the intercept on x-axis and ‘b’ is the intercept on y-axis.
The point of intersection of any two lines of the form y = ax + b and y = cx + d is same as the solution arrived at when these two equations are solved.

Perpendicular Distance:
The length of perpendicular from a given point (x1,y1)$\left(x_{1,}{y}_1\right)$ to a given line ax + by + c = 0 is ∣∣∣ax1+by1+c(a2+b2)−−−−−−−√∣∣∣=p$\left|{\displaystyle \frac{a{x}_1+b{y}_1+c}{\sqrt{(a^2+b^2)}}}\right|=p$, where p is the length of perpendicular. In particular, the length of perpendicular from origin

Distance Between two parallel lines:
Distance between two parallel straight lines ax+by+c1$ax+by+c_1$ = 0 and ax+by+c2$ax+by+c_2$ = 0 is given by |c1−c2|a2+b2−−−−−−−√${\displaystyle \frac{\left|c_1-c_2\right|}{\sqrt{a^2+b^2}}}$

Actute Angle between two lines:
If a1x+b1y+c1$a_{1}x+b_{1}y+c_1$ = 0  and a2x+b2y+c2$a_{2}x+b_{2}y+c_2$ = 0 are equations of two lines, then acute angle (q) between them is given by Cosθ=|a1a2+b1b2|(a21+b21)(a22+b22)−−−−−−−−−−−−−−−−−−−√$Cos\theta ={\displaystyle \frac{\left|a_1{a}_2+b_1{b}_2\right|}{\sqrt{\left({a^2}_1+{b^2}_1\right)\left({a^2}_2+{b^2}_2\right)}}}$

If m1$m_1$ and m2$m_2$ are slopes of two straight lines, then acute angle (θ$\theta$) between them is given by tanθ=m1−m21+m1m2$\tan \theta ={\displaystyle \frac{m_1-m_2}{1+m_1{m}_2}}$

Area of the Triangle:
The area of a triangle whose vertices are (x1,y1)$\left(x_{1,}{y}_1\right)$, (x2,y2)$\left(x_{2,}{y}_2\right)$ and (x3,y3)$\left(x_{3,}{y}_3\right)$ = 12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]