Getting started with Coordinate Geometry
Once upon a time, in a small town, a group of high school students were attending their first class in Coordinate Geometry. They were about to embark on a Mathematical journey that promised to reveal the hidden secrets of the Cartesian plane. Ms Mathews, their teacher, was known to make even the most difficult concepts simple.
They started their journey by understanding the XY plane, the coordinate axes and the origin.They learnt that any point on the xy plane could be represented by an ordered pair (x,y). Their teacher shared a story about pirates and the hidden treasures.
The x axis represented the coastline, the y axis was the depth and the treasure was hidden at different points.They learnt about lines, slopes and equations.They realised how each of these would serve as navigational tools to their treasures.
The students learnt to calculate equation of a line and their points of intersections which represented part of the hidden treasures. They realised practical applications to their study such as finding the slope of a ski park, finding the distance between two points in a field and so on. With each passing day, with consistent practice, their confidence grew.
They also learnt about famous mathematicians and scientists who had used Coordinate Geometry for their research. As days progressed, the students learnt about Circles and how to compute equation of a circle.The Cartesian plane became a canvas through which they could solve many problems while visualising them.
As the semester came to an end, the students had also developed a deeper understanding of the power of Math to solve real life problems, along with a sound knowledge of Coordinate Geometry. They realised how every point in the plane held a promise of discovery.
Would you care for an opportunity to learn Coordinate Geometry for Class 11 Mathematics? My students generally find Coordinate geometry easy. But yet, there are certain formulas and properties which you need to remember and keep in mind.
My students often get slightly confused with abscissa and ordinate. Keep in mind that abscissa stands for the x coordinate and ordinate stands for the y coordinate.
Points to remember
Remember the following:
Distance between 2 points is called the distance formula
Section formula -where a point divides the join of 2 points internally or externally in the ratio m:n. Note that there is a slight variation in the formula for external division. I have to keep reminding my students of this while problem solving.
Centroid of a triangle.
Locus of a point is the path traced by a moving point as it satisfies a given condition. As I introduce the concept of locus to my students, initially, they find it difficult to comprehend. I suppose I also faced the same problem when I started studying locus for the first time.But, not to worry, with practice it will make sense to you.
Coordinates of a point when the origin is shifted without changing the direction of the axes.
Slope of a line both using the formula and as the tangent of the angle made by the line with the positive direction of the x axis. Students find this easy, in general. Note that, when 2 lines are parallel, they have the same slope and when 2 lines are perpendicular, the product of their slopes is -1.
Finding the angle between two lines
Finding the angle between 2 lines- You need to keep in mind that when you take the modulus, it becomes finding the acute angle between the 2 lines.
Equation of a line using the slope intercept form and given 2 points. Yet another easy topic where my students usually get it right. A number of problems can be tried using this concept.
Intercept form of a line -An easy yet important formula if you are given both the x and y intercepts.
Normal form of a line- Initially when I introduce this topic, my students find it slightly difficult. But once they are able to identify the length of the perpendicular from the origin to the line and the angle made by the perpendicular with the positive direction of the x axis, it becomes easier.
Distance of a point from a line- You will use this formula in Conics and in Class 12. In this context, you should also know how to calculate distance between 2 parallel lines.
Position of 2 points relative to a line: My students just love this! Using this formula, you can determine whether 2 points lie on the same side or on opposite sides of a line.
Bisectors of angles between 2 lines: You need to note this method which is important in ISC Class 11 exams.
You need to remember three points here, firstly how to calculate the bisectors of the angles between the lines, secondly, how to calculate the bisector of the acute angle and thirdly how to calculate the bisector of the angle in which the origin lies. All three methods are interlinked and follow the same approach to start with.
I also encourage my students to learn how to calculate the circumcentre and orthocentre of the triangle. As I tell them, the orthocentre is the point of intersection of the altitudes and the circumcentre is the centre of the circumcircle of the triangle.
Circles in Coordinate Geometry
Moving on to Circles, the next topic in Coordinate Geometry for Class 11, you will start with
Equation of a circle: My students usually get this right. There are 2 forms, equation of the circle with centre at the origin and radius r and equation of a circle with centre at (h,k) and radius r.
Diameter form of a circle: This is also a formula which I encourage you to know, though you can also calculate it using the centre and radius.
General form of a circle: This is an important concept in Circles which is also used in Complex numbers while calculating the locus of a complex number. Initially, my students find it difficult, but as I proceed to do problem solving with them, they sort of, start enjoying it. There are three points you need to identify, namely, g, f and c.
There are a number of problems that can be worked around this. One of them is how to calculate the equation of a circle passing through 3 points.
In Class 10, you may have worked on calculating angles in a circle. In Class 11, you will deal with equation of a circle and problems based on these.
Condition of tangency of a circle This is the concluding part of this chapter and my students by now are able to grasp the concept of a tangent. Condition of tangency is given here. You can also calculate it using the distance formula and noting that the tangent is perpendicular to the radius at that point. As I tell my students, you keep recalling whatever you have learnt earlier in Math. So, the distance formula learnt in Straight lines can be used here.
One point which you should note is that if 2 circles touch each other externally, distance between their centres is the sum of their radii and if 2 circles touch each other internally, distance between their centres is the difference between their radii.
Summarising, I have given a brief outline of all the topics in Coordinate Geometry, namely, Straight Lines and Circles which you should be knowing.
Would you care to send this advice to hundreds of fellow students who need that extra guidance in Mathematics?
Do you need extra help?
In case you still need help, you can contact me for online tutoring.
You can choose between one to one tutoring or group tutoring. You have a choice to be tutored only in Coordinate Geometry or in any topic or subtopic of your choice.
My students have excelled in Mathematics and many of them have chosen to pursue Math as their further study. With the NEP setting in, questions tend to be out of the box. So, get that extra help.
I hope this post is useful to you. Motivating you to learn Mathematics!