COORDINATE GEOMETRY
CARTESIAN COORDINATES
A Cartesian Coordinate system (also called Rectangular Coordinate system) consists of two real number lines, one horizontal line called the x-axis and one vertical line called the y-axis. The point at which these axes intersect is called the origin. The axes divide the coordinate plane in four quadrants which are numbered I, II, III and IV as illustrated below :
Coordinates of a point: Starting at the origin, how can we reach points A,B,C and D? We have to move some units right or left and then some units up or down. To reach A we can move 1 unit to the right and 3 units up. Then the coordinates (position) of A is written as (1,3). Here 1 is the x-coordinate and 3 is the y-coordinate. X-coordinate is positive if we move right and negative if we move left. Y-coordinate is positive if we move up and negative if we move down. Thus the coordinates of B(-2,1), C(-1,-2) and D(3,-1). The x-coordinate is also called abscissa and the y-coordinate is called ordinate.
The notation P(x,y) means P is a point whose coordinates are (x,y). In cartesian coordinate system we can draw graphs of functions (refer to the chapter Function - Graphs of Functions). The functions give the graphs of lines, circles, parabola, hyperbola, ellipse and many other curves.
LINES AND SEGMENTS
Distance between two points P x1 , y1 and Q x 2 , y 2 are given by
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d = q x 2 @ x1 + y 2 @ y1
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The coordinates of the mid point of the line segment joining P x1 , y1 and Q x 2 , y 2 is f xfffxff yffffff + ff 1 +ff g fffff ffffff fff2 f fffy 2f f 1f ff , ff ff 2 2 Example : Find the distance between the points A (1,-3) and B (-2,1) and also the midpoint of the segment AB. Solution : Taking x1 = 1, y1 = @ 3, x 2 = @ 2, y 2 = 1
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2 1 2 1
+ xff ff2f xfffff 1ffff 1f @f ffff ffff f f 1f ff The x @ coordinate of the mid point = ffff2f= ffff= @ f 2 2 2 + ff f3 ffff 2f yffffff @fffff ffffff ffffff fffy 2f ffffff ffff f+ 1 f f and the y @ coordinate = 1 = =@ =@1 2 2 2 f g 1f f So the mid point is @ f, @ 1 2
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SLOPE OF A LINE
The slope of a line is a number which indicates the direction or the slant of the line. It is generally represented by “m”. For a non-vertical line passing through ` a ` a P x1 , y1 and Q x 2 , y 2 the slope of the line segment PQ is yffffff Δyf ffffff fffff ff ff f 2 @ y1 Slope = m = ffffff = fff = x 2 @ x1 Δx Changefffffffffffffff fffffffffffffffffffff fffffffffffcoordinatef fffffffffffffffffff fffff in y @ fffffffff Change in x @ coordinate
For any line the slope is fixed (or unique) and it is independent of the points selected. In 3ffff 1f @f ffff f ff1f the picture the slope of the line calculated taking B(2,1) and C (6,3) is m = ffff= f , 6@2 2 ` a 3fff@fff 1f @ fff fffffff f ffff2 f ` a which is the same as taking A(-4,-2) and C(6,3) m = fffffff= f . 6@ @4 2
The concept of slope is very important in understanding various concepts of Calculus like derivatives, maxima, minima etc. We have shown below slopes of some lines.
The slope of a horizontal line is 0, because as we move along the line the x-coordinate Δyf 0f ff ff ff ff f changes, but the y coordinate does not change. Δy = 0 here A So, m = fff fff 0 A = f= Δx Δx
The slope of a vertical line is not defined, because as we move along the line the xΔyf Δyf ff ff ff ff f coordinate does not change, but the y coordinate changes. Δx = 0 here A So, m = fff fff = f Δx 0 which is not defined (a symbol with denominator 0 does not signify a number). For other lines, if the slope is positive, then it slopes up i.e. with increase in xcoordinate, the y-coordinate increases. If the slope is negative, then it slopes down i.e. with increase in x-coordinate, the y-coordinate decreases.
Parallel & Perpendicular lines:
If two lines l 1 and l 2 have slopes m1 and m2 : The lines are parallel or l1 || l 2 if and only if m1 = m2 The lines are perpendicular or l1 ? l 2 if and only if m1 Bm2 = @ 1 In the above picture the lines l 3 ? l 4 as their slopes are 1 and -1 and ` a m1 B m2 = 1 A @ 1 = @ 1
Though vertical lines have no slopes, when two lines are vertical (i.e. both of them have no slopes), we know they are parallel. Similarly two horizontal lines are parallel.
We know that a vertical line and a horizontal line are perpendicular to each other, so a line with no slope and another line with 0 slope are perpendicular to each other. Thus, in the above picture the lines l 2 and l1 are perpendicular. Example : Find the slope of the line joining points A (1,-3) and B (-2,1). Solution : Taking x1 = 1, y1 = @ 3, x 2 = @ 2, y 2 = 1 ` a yffffff 1fff@fff fff @ fff 4f ffffff fffffff fff ffffff fffffff 4ff fffff ffff3 f ff f f 2 @ y1 =` a = f =@ f Slope of AB = m = 3 x 2 @ x1 @2 @1 @3
Example : Find the slope of a line perpendicular to the line joining A(-3,1) and B(2,3). Solution :
Slope of AB = m1 = 2@ @3 5 If the slope of the line perpendicular to it is m2 then m1 Bm2 = @ 1 5f 2f f f f f multiply both sides with [ Bm2 = @ 1 2 5 5f 5f f f f f [ m2 = @ A # Slope of a line perpendicular to AB is @ A 2 2 3@1 fffffff 2f fffffff f fffffff f fffffff ` a=
DIFFERENT FORMS OF EQUATIONS FOR STRAIGHT LINES General form : A linear equation of x and y represents a straight line and is called the general form of the equation of the line. This is written as
Ax + By + C = 0 where A,B,C are real numbers and both A and B are not 0 at the same time Thus 2x @ 3y @ 5 = 0, p3 x + y = 0, 3x @ 2 = 0, y @ 4 = 0 are equations of straight lines. In 2x @ 3y @ 5 = 0, A = 2, B = @ 3, C = @ 5 A w w w w w w w w w w w w In p3 x + y = 0, A = p3 , B = 1, C = 0 A
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Many a times we are required to find the general form of equation of a line from its slope, a few known points on the line or its intercepts on the axes. There we take help of the following forms :
Point-Slope form : ` a If the line passes through the point x1 , y1 and its slope is m , then the equation of the line can be written in the form
y @ y1 = m x @ x1
` a
Intercept Form : If a line intersects the x-axis at A(a,0) and the y-axis at B(0,b) then a is called the xintercept and b the y-intercept. Then the equation of the line can be written in the form
xf yf f ff f f f f + = 1 where a ≠ 0, b ≠ 0 a b
Slope-Intercept form : If the slope of a line is m and it intersects the y-axis at B(0,b) ( i.e. the y-intercept is b ), then the equation is y = mx + b
Distance between the point and a line : The distance between the point and a line is the shortest distance a between them. This is ` A x1 , y1 , the distance of this the perpendicular distance between them. For a point point from the line Ax + By + C = 0 is given by L
It can also be written in another form. The distance of the point A x1 , y1 from the line with equation y = mx + b is M L M L Lmxfffffffff b@ M ffffffffff ffffffffff f1 +fff y1 M L ffffwwfff wwww wwww wwww wwww www www w d =L wwww M M L q L 1 + m2 M
M L M L Axffffffffff +fffffff ByfffCf +fff ffffffff M ffffff1 f1 f f L ffwwww f M wwwww wwww wwww wwww wwww wwww d =L wwwww M L q 2 M 2 L A +B M
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Example : If the equation of a straight line is 2x-3y-6 = 0 , find the a) Slope b) x-intercept
c) y-intercept d) distance of the point (4,8) from this line Solution : 2x @ 3y @ 6 = 0 adding @ 2x + 6 on both sides @ 3y = @ 2x + 6 dividing both sides by @ 3 2f f y = fx @ 2 3 comparing this with the slope @ intercept form we get ` a 2f f a slope m = f and 3 ` a c y @ intercept b = @ 2 putting y = 0 in the general form we get ` a 2x @ 3 0 @ 6 = 0 [ 2x = 6 [ x = 3 the line intersects the x @ axis at 2,0 b so x @ intercept = 3
b c b c
` a
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d distance of the point 4,8 from this line is
M L L M M L M L2.4fffffffff L@fff ffff 22M 22 f ffffffffff L ffff ffff ffffffffff f @ fffffff f3.8 @ M ffff ffff ff M ff L fffwwww6M L fwf wwwwww M=L wwM= ww wwwwww wwwwww wwwwww wwwwww wwwww wwwww w ww ww w w w w ww ww w w w w w d =L a2 M Lp13 M p13 Lq 2 ` M L
2 + @3
SYMMETRY
Two points A and B are symmetric with respect to the point Q, if Q is the mid point of the line segment AB. Two points A and B are called symmetric with respect to line l, if l is the perpendicular bisector of the line segment AB. A figure is symmetric with respect to a point Q if for every point P1 on the figure there is another point P2 on the figure such that P1 and P2 are symmetric relative to that point Q. Similarly, a figure is symmetric with respect to a line l if for every point P1 on the figure there is another point P2 on the figure such that P1 and P2 are symmetric with respect to that line l . When drawing the graphs of equations, it is often useful to know if the figure is symmetric with respect to a point or line. A graph is symmetric with respect to the • y-axis, if (-a,b) is on the graph whenever (a,b) is on the graph • x-axis, if (a,-b) is on the graph whenever (a,b) is on the graph
• •
origin(0,0), if (-a,-b) is on the graph whenever (a,b) is on the graph line y=x, if (b,a) is on the graph whenever (a,b) is on the graph.
Tests of symmetry : 1. If by substituting x with –x in the given equation we get the same equation, it is symmetric with respect to the y-axis. 2. If by substituting y with –y in the given equation we get the same equation, it is symmetric with respect to the x-axis. 3. If by substituting x with –x and y with –y simultaneously in the given equation we get the same equation, it is symmetric with respect to the origin. 4. If by interchanging x with y in the given equation we get the same equation, it is symmetric with respect to the line y=x.
Example : Test the equation y 2 = x + 3 for symmetry and draw the graph. Solution : We perform all the tests of symmetry on this equation. The equation remains unchanged when y is replaced with –y. So the equation is symmetric with respect to x. We get the following points in the 1st quadrant and then reflect the graph through the x axis. -3 -2 1 6 13 x: 0 1 2 3 4 y:
INTERCEPTS AND ASYMPTOTES
The x-intercept and y-intercept of straight lines were discussed earlier. These can similarly be applied to any graphs. Unlike straight lines other graphs may intersect the x-
axis and y-axis at more than one points. The x-coordinates of all the points where the curve intersects the x-axis are called the x-intercepts. The y-coordinates of all the points where the curve intersects the y-axis are called the y-intercepts.
In the picture the x-intercepts for the curve are x1 , x 2 and x 3 and the y-intercept is y1 . When the graph of an equation approaches a line but never touches it, then the line is called an asymptote to the curve. In the graphs of the hyperbolas given in the chapter of Conic Sections, the asymptotes are shown.
PARAMETRIC EQUATIONS
Generally we describe a curve by giving a direct relationship between the variables x and y. However, sometimes it is useful to introduce a third variable (say t or θ ) and express x and y in terms of the third variable. The third variable is called a parameter and the equations are called parametric equations. For example, the equation of the parabola 4y = x 2 and the the parametric equation x = 2t,y = t 2 are equivalent. Irrespective of whatever value we choose for t, it will satisfy the equation 4y = x 2 . If we choose t=1, we get x=2.1=2 and y=1 i.e. we get the point (2,1) which is on the curve 4y = x 2 .
POLAR COORDINATES
Besides cartesian coordinate system, there is another coordinate system called the Polar jj jj jj j j j j k j Coordinate system. In this a fixed point O is called the pole and a fixed ray OA starting
from the pole is called the polar axis. The polar axis is drawn horizontally to the right side of the pole as shown below. To assign the coordinates to any point P lying on the plane, P is joined to O by the ray jk jj jj jj j j j j j jk jj jj jj j j j j j jj jj jj j j j j k j OP . If θ is the angle the ray OP forms with OA , and the distance OP = r, then the polar coordinates of P is written as (r, θ ).
ray In case r is negative in (r, θ ) measure |r| units along the d directed opposite to the e πf ff f f terminal side of θ . For example in the polar coordinates @ 5, , the ray directed 3 2ff πf πf f f ff f opposite to the terminal side of the angle ffis the terminal side of the angle ff. So the 3 3 f g 2ff πf ff ff f . point is same as 5, 3 Any polar coordinate with r = 0 represents the pole.
Transformation Relationship with Cartesian Coordinates :
If the point P has polar coordinates (r, θ ) and Cartesian coordinates (x,y) then
x = r cosθ and y = r sinθ r2 = x 2 + y2 a yf ` f f tanθ = ff x ≠ 0 x
2πf ff ff f Example : Convert (4, ff ) to Cartesian coordinates. 3 Solution :
2πf ff ff f Here r = 4,θ = ff, using transformation formula 3 f g 1f 2πf ff ff ff f f f = 4 @ =@2 x = r cosθ = 4 cos 2 3
w w w w w f pwg w w w w w w 3 2πf ff ff ff f fff fff fff ff =4 = 2 p3 y = r sinθ = 4 sin 2 3 b
So the cartesian coordinates are @ 2,2 p3
wc w w w w w
Example : Convert (2, -2) to polar coordinates with r>0 and 0 ≤ θ ≤ 2π . Solution : a2 2 ` Since x = 2 and y = @ 2 r 2 = x 2 + y 2 = 2 + @ 2 = 4 + 4 = 8 As r>0, we take the positive square root and r = 2 p2 Since the point lies in quadrant IV and yf @ ff 7ff πf f f2f f ff ff f tanθ = ff fff= @ 1 [ θ = fff = ff x 2 4 f w w 7ff w ff w w πf w fg So the polar coordinates are 2 p2 , fff 4
w w w w w w
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