# Straight Lines: JEE Main Maths Topics to Study Straight Lines JEE is the foundation of Coordinate Geometry. Coordinate Geometry is that part of JEE Mathematics which almost covers 20% weightage of JEE Advanced Mathematics Paper. This tells us how important Straight Lines JEE is.

Some previously learned concepts that we must know:

• The distance between two points if coordinates are given
• Section formula
• Area of a triangle if coordinates of three points are given and hence the condition of collinearity of three points
• Coordinates of special points like: Centroid, Incenter, Circumcenter, Excenter of a triangle

To plot any curve, we should have the basic knowledge about the locus of a point, i.e. what constraints the point is being put under, what conditions it should follow. A straight line is the locus of a point which is collinear with two given points.

## Important Terms

To study about a straight line you should be well acquainted with terms like:

• Slope of a line: You should be well acquainted with geometrical as well as the algebraic definition of the slope of the line. This is important as far as Straight Lines JEE is concerned.

Note: Slope of vertical line(parallel to y-axis) is not defined, whereas slope of a horizontal line(parallel to x-axis) is zero.

• The Intercept of a line: Generally, students consider “Length of an intercept” and “Intercept cut by a line” as the same. But there is a big difference between the two, and JEE has also asked a question on this concept previously.

Point to be remembered: Length of an intercept cut is always non-negative whereas Intercept cut by a line can have any Real value. Suppose if in a question you are given that a line has a slope of 1 and has “length of intercept” equals to 1, then there will be two such lines(y=x+1,y=x-1). But if it is said that intercept of the line is 1, then there will be only one such line i.e (y=x+1).

• The angle between two lines: One very important point that needs to be kept in mind is that angle is always measured in an anti-clockwise sense.
• Mathematical conditions for two lines to be parallel/perpendicular.
• Various Forms of Equations of a line: This is a very important portion for Straight Lines JEE. You should be thorough with each and every form. Especially the Parametric form, Normal form and Intercept form. JEE has been asking questions on these form constantly. The best part of these forms being that they sometimes reduce very complex questions to a simple form.
• The general form of the equation of a line: When nothing clicks about a question, this form of the line helps us. You may assume a general line and then apply the given boundary conditions.
• The distance of a point from a line: Here we always talk about the perpendicular/shortest distance.
• Distance between two parallel lines

This was all about general terms related to Straight Lines JEE. Now we will be covering sub-topics of Straight Lines JEE which are important.

## Important Topics

• Position of a point w.r.t line:

We can say that the point A(x1,y1) is above the line ax+by+c=0 if

1. ax1+by1+c>0 and b>0
2. ax1+by1+c<0 and b<0

We conclude that the point A(x1,y1) is below the line ax+by+c=0 if

1. ax1+by1+c<0 and b>0
2. ax1+by1+c>0 and b<0

Note: The point A(x1,y1) will be on the line ax+by+c=0 if ax1+by1+c=0.
You should remember this formula otherwise the process is sometimes too lengthy.

• Image of a point in a Line:

There is no need to cram formula for this. We use two basic concepts for this problem.

1. The condition of slope (since given line and the line joining the given point and its mirror image are perpendicular to each other).
2. The midpoint of the line segment joining point and it’s mirror image lie on given line.
• Foot of Perpendicular:

In this case one can say that image itself lies on the given line. So that point satisfies the equation of given line.
Another equation can be derived from condition of slope.

• Angle Bisector of the pair of straight lines:
This is one of the most important parts of Straight Lines JEE. Many aspirants face problem in this part. This is a bit lengthy and confusing as well. Don’t worry, we will make out some tricks.

Of course, there are two angle bisectors given by the equation: How can we differentiate between the acute angle bisector and the obtuse angle bisector?

It is very simple to determine whether the bisector is an acute angle bisector or an obtuse angle bisector. Compute the angle between the initial line and one of the bisectors. Suppose that angle is θ. Then find the tangent of this angle θ.

In case |tan θ| < 1, then we have 2θ < 90, then this represents the obtuse angle bisector.
On the other hand, if |tan θ| > 1, then this represents the obtuse angle bisector.

It is highly recommended that you should learn the direct results written below:

1. When both c1 and c2 are of the same sign, evaluate a1a2 + b1b2. If negative, then acute angle bisector is (a1x + b1y + c1)/√(a12 + b12) = + (a2x + b2y + c2)/√(a22 + b22).
2. When both c1 and c2 are of the same sign, the equation of the bisector of the angle which contains the origin is (a1x + b1y + c1)/√(a12 + b12) = + (a2x + b2y + c2)/√(a22 + b22).
3. Bisectors of the angle containing the point (α, ß) is (a1x + b1y + c1)/√(a12 + b12) = + (a2x + b2y + c2)/√(a22 + b22) if a1α + b1ß + c1 and a2α + b2ß + c2 have the same sign.
4. Bisectors of the angle containing the point (α, ß) is (a1x + b1y + c1)/√(a12 + b12) = + (a2x + b2y + c2)/√(a22 + b22) if a1α + b1ß + c1 and a2α + b2ß + c2 have the opposite sign.
• Rotation of a line:

This concept is used in many problems directly. Rotation of line is also helpful in Optics.
Rotation by θ radians counterclockwise, around the origin gives the new coordinates as:
x=ucosθ+vsinθ
y=−usinθ+vcosθ

• Pair of Straight Lines:
This portion of Straight Lines JEE is also important. Here we also use Determinants for finding out different conditions. JEE always love interlinking topics, hence making it important. The concept of Homogenisation is of prime importance and is being constantly asked.
• The concept of the family of straight lines:
This is an easy concept which is very very useful in solving problems.
The general equation of the family of lines through the point of intersection of two given lines is L + λL’ = 0, where L = 0 and L’ = 0 are the two given lines, and λ is a parameter.

These are the important points, tricks which you should keep in mind while solving Straight Lines JEE.

The key to score high in Coordinate Geometry is Practice, Practice and Practice, along with a number of direct results you can remember. You should focus on these points.

The textbook that you can refer for Straight Lines JEE is S.K Goyal. It contains a nice set of questions. One should always attempt previous years’ questions of JEE Main and JEE Advanced. If you are able to solve them by yourself, it will boost your confidence a lot.

Practice hard and we are sure that you will come up with flying colors. There is no shortcut to success. Remember — hard work beats talent, EVERY SINGLE TIME. The more questions you solve, the better you will get at solving questions — which has a direct effect on your JEE rank.