Co-ordinate Geometry

Lesson: Equation of lines

Prakash AdhikariIslington college, Kamalpokhari Kathmandu

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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

Lesson Description

Warm up questions

Graph of Power of x

Simultaneous Equations

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

Graph of Power of x ( Board work)

• y = x-2

• With Fractional powers of x

for 0<n<1

1

2y x

133y x x

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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Thank You for your active partication

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