Sunday, December 17

Co-Ordinate Geometry

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In a Catesian co-ordinate system a position in 2 diamension is given by reference to a origin in horizontal and vertical direction. The north and east of the refernce axis or y axis and x axis is positive and South and West from the y axis and x axis is negative directions. In other words, it can express relationship between two variables called independent and dependent variables. That is Co-ordinates represents the ordered pairs of two variables. In this respect The co-ordiate system can be a graphical representation of the relationship between two variable the the way it is  related between certain domain of the x variables and the range of values it can have. For example the Co-ordinate system is a useful method to to solve inequalities or to find the roots of a equation or to verify whether it has real roots which satisfy the equation.

If  ay + bx + c = 0 and if the x is to the power of one, then it can be represented in co-ordinate system as a relationship between x variable and the y variable. In this instant it is a straight line. If one re-arrange the above equation this will become as follows:

ay = -bx -c

y = -b/a x – c/a.  when x = 0 then y = -b/a. This is called a y intercept. The rate of increase of y variable to that of the increase in x variable or change in x variable is called gradient. That is, given the narure of the above equation the gradient for any domain of x variable the change in y variable is a constant. In this instant, the gradient say (m) is equal to -b/a. That is, the straight line equation can be written as y = mx + C, If m is greater than 0 for any increase in x variable the the y variable will increase, therefore the straight line will sllope upwards. If m<0, then the straight line will slope downwards.

That is given the equation, one can predict the nature of the linear relationship between two variables. If it is represnted in a Cartesian Co-ordinate system it will portray the points less than and greater than y as well, it also knows the values where the y values less than 0 and greater than 0 from the x intercept. In other words, Co-ordinate geometry can be helpful to solve inequalities or in solving algebra problems. In addition, as discussed above it can also be useful in Analytical geometry and discover the properties of certain geometrical figures or to prove some geometric Properties are correct.

Example 1

If y = -2x + 3, find the range of y values less than 0 and more than 0?

Solution 1

As m <0, the straight line will slope downwards. If one draws a graph given m = -2 and y intercept as 3 one can observe that the line cuts the x axis at 3/2. One can verify this by Alebra as 0 = -2x +3 and there fore x = 3/2. As the straight line is sloping down wards, If x > 3/2 the y values will be less than 0 as dipicted by the graph as well if x, 3/2 the y values are greater than 0. This demonstrate the usefulness of strudying any mathematical relationsip. In nature most variables with some randomness behave in this manner. In this respect, it can be usefull in spacial problems, optimizing problems and in studying the properties of various graphs as locus.

Co-ordinate Geometry and quardratic inequalities

As discussed above, A fuction or relation is a linear function only if the highest power of the x variable is 1. If the the power is greater than 1, then the function or relation become a continous or discontinuous curves or graphs having many roots which is real or complex depending on the co-efficents of x in any polynomial function. Quardratic function is a subset of polynomial function. It is ithe form as follows:

y = a* x squared + bx + C.

If a > 0, If x > 0 then the function increases as x increases. As well, if one plots for given values of x the y values it will decrease when x is less 0 and increase when x is greater than 0. In other words, this is a curve. As well it is a special curve called parabola. That is if a >0, the quardratic function will have a minimum point. There fore it a<0, then the above equation will have a maximum point. There fore this can be useful for solving soem optimization problems. The values of a, b and c will determine horizontal vertical and the shape of the curve whether it is more closer to the minimum or maximum points or a flatter curve. In adddition, it will also determine whether it cuts the x axis. If it cuts the x axis, then between certain domain of x values the function will be positive or negative depending whether the curve, curves upwards or down wards. That is, if a > 0, then the function will be positive between the co-ordinates where the curve cuts the x axis. If a <0, then the function is negative between the co-ordinates where the the curve cuts the x axis. If b squared – 4*ac is greater than 0 it will cut the x axis if it is less than 0 then the curve will be up or down depending whether a is greater or less than 0.  Outside the the roots of the function the the values will be opposite to that of the range of values it may have between the curves intersection of the x axis. In other words, if the y values are negative between the roots of the function then the values outside the roots are positive. If they are positive, then they will be negative outside the roots of the function.

This illustrates how  Co-Odinate geometry can be an analytical tool to solve quardratice inequalities and to solve optimization problems. The xample below will illustrate how Co-ordinate gGeometry can help to solve quardratic inequalities.

Example

If y = – 3x squared –  7x + 3 find the range of values where y is less than 0 ?

Solution

First draw the graph for some variables of x. One can see as b squared –  4* ac in this instant is greater than 0, It will cut the x axis. If y = 0 then – 3 x squared – 7x + 3 = 0. Solving this equation using the quardratic formula one gets – (-7) + or – square root of (-7) squared – 4* (-3)* 3 = 7 + or – square root of ( 49 + 36) = 7 + or – squaroot of 85. That is the curve will cut the x axis between 7 + square root of 85 and 7 – square root of 85. In this case a is less than 0. There fore between these two points the y is greater than 0  That is y is less than 0 when x isless than 7 – square root of 85 and x is greater than 7 + square root of 85. 

Co-ordinate geometry and Locus of a moving point

Say q pint is moving from the co-ordinates say (f, 0) and ( -f,0) and the moving of that point is obeyed by a law of the addition of the distances from the focus is equal to the total distances of the moving point which is a constant equal to the total distance from the origin. Say the half of the horizontal semi axis is (a) and minor axis is (b). In this instant b squared + f squared = a squared by taking the point (0, b).

Now applying the Locus definition one can derive the equation from the co-ordinate gemetry principle of the distance between two pints is given by the square root of th squared distance addition of x co-ordinate difference . Applying this concept, the following equation of ellipse can be worked out as follows:

x squared/ a sqaured + y squared / b squared = 1, If, a is a major semis axis and b is a minor semi axis and the origin is (0,0). If the major axis is b and minor axis is a, and the the focus is in the y axis, the ellipse formula is as follows:

x squared / b squared + y squared/ a squared = 1.

One can see from the above the usefulness of Co-ordinate geometry to analyse the properties fifferent functions and relations and these exist in nature. For example in Splar system most of the Planetary orbits are an ellipse as sun in one of the focus of the orbit.

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