co-ordinate geometry lesson: equation of lines prakash adhikari islington college, kamalpokhari...
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Co-ordinate Geometry
Lesson: Equation of lines
Prakash AdhikariIslington college, Kamalpokhari Kathmandu
1
Review of Last Lecture
Equation of straight line in the form of y=mx+c
Gradient , x and y intercept of straight line
Equation of straight line with two end points
General equation of straight line in the form of ax+by+c=0
Warm up questions• What is the Equation of straight lines in
– General form?– slope intercept form?– Parallel to x axis?– Parallel to y axis?– Point Slope form?
– Two points (x1,y1) and (x2,y2) ?
• when you don’t have a y-intercept?
x
y
• How can you tell any point lie on the line?• Ex : the points A(7,0) and B(2,5) line on the line y= x-7?• Ex : the point (2,4) and (1,5) lies on the curve y=3x2+2?
Warm up questions
Warm up questions
• Does c carries same meaning for both form equation of straight lines y=mx+c and ax+by+c=0?
Graph of Power function of x ( Board work)
• With positive Powers• Straight line y=x index n=1• Parabola y = x2
index n>1• Cubic function y = x3
• With negative powers of x• Hyperbola y = x-1
Index n<0
Point of intersection of two lines
• If two straight lines are 4x-6y=-4 and 8x+2y=48 Q. How do you find the point of intersection of these lines?
• You want a common point (x,y)
which lies on both the lines so need to
solve the equation simultaneously. • Note:
This argument applies to the straight lines
with any equation they are not parallel.
Simultaneous Equations and Intersections
We will deal with• Intersection of – two straight lines– One straight line and a curve– Two curves
1. Suppose we want to find the common point where two lines meet.
The point of intersection has an x-value 2 and y-value 1.
Ex 1. x = 2y and y = 2x-3
When Sketching the lines with plotting points gives
The point of intersection has an x-value between -1 and 0 and a y-value between 3 and 4.
Ex 2. y=-x+3 and y=2x+5
Sketching the lines gives
The exact values can be found by solving the equations simultaneously
x32
As the y-values are the same, the right-hand sides of the equations must also be the same.
3 xy52 xy
At the point of intersection, we notice that the x-values on both lines are the same and the y-values are also the same.
yy
523 xx
Substituting into one of the original equations, we can find y:3 xy
332 y
311y
The point of intersection is 311
32 ,
x 32
311
32 ,
Sometimes the equations first need to be rearranged:
155 x
Substituting into (1):
The point of intersection is ),( 23
3x
Solution: Equation (2) can be written as
Now, eliminating y between (1) and (2a) gives:
xy 311 )( a2
xx 31142
42 xy
113 yx
)(1
)(2e.g. 2
42 xy 2y
There are 2 points of intersection
2xy
We again solve the equations simultaneously but this time there will be 2 pairs of x- and y-values
xy 23
Ex2. Find the points of intersection of and xy 23
2xy
Since the y-values are equal we can eliminate y by equating the right hand sides of the equations:
Another way,
xy 23 )(2
2xy )(1
xx 232 This is a quadratic equation, so we get zero on one side and try to factorise:0322 xx
031 ))(( xx 31 xx or
To find the y-values, we use the linear equation, which in this example is equation (2)
11231 yyx )(
93233 yyx )(
The points of intersection are (1, 1) and (-3, 9)
32 xy
13 xy
)2,1(
)13,4(
13 xy )(2
32 xy )(1
Ex. 3 Sometimes we need to rearrange the linear equation before eliminating y
Rearranging (2) gives 13 xy )2( a
Eliminating y: 1332 xx0432 xx
0)4)(1( xx
1x 4xor
Substituting in (2a): 21 yx134 yx
Ex 4. Your turnFind the points of intersections of the following curve and line graphically and substitution method
8 yx )2(22 xy )(1
Solving the equations simultaneously will not give any real solutions
Special Casesex 1. Consider the following equations:
1 xy )(222 xy )(1
The line and the curve don’t meet.1 xy
22 xy
042 acb The quadratic equation has no real roots.
we try to solve the equations simultaneously:
1 xy )(222 xy )(1
Eliminate y: 122 xx
012 xx
Calculating the discriminant, we get: acb 42
))(()( 11414 22 acb
41 3 0
14 xy
32 xy
Ex 2. 14 xy )(232 xy )(1
Eliminate y: 1432 xx
The discriminant, 0)4)(1(444 22 acb0442 xx
0)2)(2( xx
(twice)2 x
The quadratic equation has equal roots.
The line is a tangent to the curve.
72 yx
0442 xxSolving
To solve a linear and a quadratic equation simultaneously:
SUMMARY
• Solve for the 2nd unknown
• Substitute into the linear equation to find the values of the 1st unknown.
2 points of intersection042 acb
the line is a tangent to the curve042 acb
042 acb the line and curve do not meet and the equations have no real solutions.
• Eliminate one unknown to give a quadratic equation in the 2nd unknown, e.g. 02 cbxax
ExercisesDecide whether the following pairs of lines and curves meet. If they do, find the point(s) of intersection. For each pair, sketch the curve and line.
1.
2.
3.
22
32
xy
xy
77
32
xy
xy
01
32
xy
xy
Solutions 2232 xx
0122 xx
0)1)(1(4442 acb
0122 xx
0)1)(1( xx1x
4y
1.
22
32
xy
xy
the line is a tangent to the curve 042 acb
32 xy
22 xy
32 xy
77 xy
Solutions 7732 xx
01072 xx
9)10)(1(44942 acb
01072 xx
0)5)(2( xx5,2x
72 yx
77
32
xy
xy2.
there are 2 points of intersection 042 acb
285 yx
32 xy
01 xy
Solutions 132 xx
042 xx
15)4)(1(4)1(4 22 acb
01
32
xy
xy3.
there are NO points of intersection 042 acb
For tutorial class
• Practice the lessons at home: Book : Pure Mathematics P1
Exercises: 3B, 3D and 3E
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