Co-ordinate Geometry: Theory & Formulas

A coordinate graph consists of a rectangular grid with two crossing lines called axes. The x-axis is the horizontal line and the y-axis is the vertical line. The axes intersect each other at the point (0,0) which is also called the origin.
Operations that can be done in Coordinate Geometry
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If we are given the coordinates of some points we can:
  • Calculate  the distance between them
  • Finding the midpoint, slope or equation of a line segment
  • Determine if the given set of lines are parallel or perpendicular
  • Finding the area or perimeter of a given polygon defined by the points..
  • Define the equations of different geometric figures.
For 2 lines having equations:
a1x+b1y=c1
a2x+b2y=c2
Unique Solution (meet at a single point) = (a1 / a2) ≠ (b1 / b2)
No solutions (does not meet) = (a1 / a2) = (b1 / b2) ≠ (c1 / c2)
Infinitely many solution (overlapping) = (a1 / a2) = (b1 / b2) = (c1 / c2)
Given below is a graph showing which coordinates are having what signs in different quadrants?
1. Distance Formula between A (x1, y1) & B (x2, y2).
Example: Find the distance between two Points: A (2,3) and B(5,6)?
Using the above formula:
The answer comes out to be 3&redic;2.
2. The coordinates of C dividing the line segment joining the points (x1,y1) & (x2,y2) internally in the ratio
m1:m2 are

Example .Find the co-ordinates of point Z, which divides the join of P (4, -5) and Q (6, 3) internally in the ratio 2 :5
Sol: Let the co-ordinates of point Z are (x, y).
3. External Division
4. Equal ratio

Example: Find the co-ordinates of the mid-point of the line segment joining the points M (4, -6) and N (-2, 4).
Different forms associated with the equations:
1. The point slope form: The equation of a straight line passing through the point.(x1,y1) and having slope m is (y-y1)=m(x-x1)
2. Two point form:
3. Slope intercepts form: The equation of a line having slope m & making an intercept C on y axis is : y=mx+C
Example: Find the equation of the line parallel to the line passing through (5, 7) and (2, 3) and having x intercept as -4.
Sol: Slope of the given line = (7 - 3) / (5 - 2) = 4/3. So the slope of the required line is also 4/3. One point on this line is (-4, 0). Hence the equation of the line is:
y - 0 = 4/3 (x + 4) ⇒ 3y = 4x +16.
4. Double intercept form :
(x/a) + (y/b) = 1
5. General form of an equation is Ax+By+C=0
6. For Parallel lines(slopes are equal) i.e. m1 =m2
7. For perpendicular Lines(product of slopes = -1) m1*m2 =-1
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Example:
What is the relation between different set of given lines:
a) y = 3x + 1
b) y = -1/3 x + 2
c) 1/3y = x - 3
The gradients of the lines are 3, -1/3 and 3 respectively. Therefore (a) and (b) and perpendicular, (b) and (c) are perpendicular and (a) and (c) are parallel.