Co-ordinate Geometry Lesson: Equation of lines Prakash Adhikari Islington 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

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 (x 1,y 1 ) and (x 2,y 2 ) ? 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=3x 2 +2? 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 Parabolay = x 2 index n>1 Cubic functiony = x 3 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

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

As the y-values are the same, the right-hand sides of the equations must also be the same. 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. Substituting into one of the original equations, we can find y: The point of intersection is

Sometimes the equations first need to be rearranged: Substituting into (1): The point of intersection is Solution: Equation (2) can be written as Now, eliminating y between (1) and (2a) gives: e.g. 2

There are 2 points of intersection We again solve the equations simultaneously but this time there will be 2 pairs of x- and y- values Ex2. Find the points of intersection of and

Since the y -values are equal we can eliminate y by equating the right hand sides of the equations: Another way, This is a quadratic equation, so we get zero on one side and try to factorise: To find the y- values, we use the linear equation, which in this example is equation (2) The points of intersection are (1, 1) and (-3, 9)

Ex. 3 Sometimes we need to rearrange the linear equation before eliminating y Rearranging (2) gives Eliminating y : or Substituting in (2a):

Ex 4. Your turn Find the points of intersections of the following curve and line graphically and substitution method

Solving the equations simultaneously will not give any real solutions Special Cases ex 1. Consider the following equations: The line and the curve don’t meet.

The quadratic equation has no real roots. we try to solve the equations simultaneously: Eliminate y: Calculating the discriminant, we get:

Ex 2. Eliminate y: The discriminant, The quadratic equation has equal roots. The line is a tangent to the curve. Solving

 To solve a linear and a quadratic equation simultaneously: SUMMARY Solve for the 2 nd unknown Substitute into the linear equation to find the values of the 1 st unknown. 2 points of intersection the line is a tangent to the curve the line and curve do not meet and the equations have no real solutions. Eliminate one unknown to give a quadratic equation in the 2 nd unknown, e.g.

Exercises Decide 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

Solutions 1. the line is a tangent to the curve

Solutions 2. there are 2 points of intersection

Solutions 3. there are NO points of intersection

For tutorial class Practice the lessons at home: Book : Pure Mathematics P1 Exercises: 3B, 3D and 3E 27

Thank You for your active partication 28