Problem 13 - Algebra for JEE ADVANCED

Question

There are \(n\) points in a plane, in which no three are in a straight line except ' \(m\) ' which are all in a straight line. Find the number of (a) different straight lines, (b) different triangles, (c) different quadrilaterals that can be formed with the given points as vertices.

Step-by-Step Solution

Answer

(a) \(\binom{n}{2} - \binom{m}{2} + 1\), (b) \(\binom{n}{3} - \binom{m}{3}\), (c) \(\binom{n}{4} - \binom{m}{4}\).

Step 1: Calculation of Total Possible Lines

To calculate the total number of possible lines that can be formed with \(n\) points, we use the combination formula for selecting 2 points out of \(n\): \(\binom{n}{2}\). This gives the total number of two-point combinations which determine lines.

Step 2: Excluding Collinear Points in Line Calculation

Since \(m\) points are collinear (meaning they are on the same straight line), they account for only one line despite being chosen in different pairs. Thus, we must subtract the extra lines counted which are only internal to this group. Therefore, the number of overlapping lines is \(\binom{m}{2}\).

Step 3: Total Number of Different Lines

The number of different lines is the difference between the total possible lines and the overlapping lines formed by the collinear points. Hence, the number of different lines = \(\binom{n}{2} - \binom{m}{2} + 1\).

Step 4: Total Possible Triangles

To find the number of triangles, we need to select any 3 points from \(n\). This can be calculated using the combination \(\binom{n}{3}\).

Step 5: Excluding Degenerate Triangles with Collinear Points

Any triangle using all 3 of the collinear \(m\) points would be degenerate (they lie on a straight line). Therefore, we must exclude these cases: \(\binom{m}{3}\).

Step 6: Total Number of Different Triangles

The number of different triangles that can be formed is therefore given by \(\binom{n}{3} - \binom{m}{3}\).

Step 7: Total Possible Quadrilaterals

To find the number of quadrilaterals, we need to select any 4 points from \(n\). This can be calculated using the combination \(\binom{n}{4}\).

Step 8: Excluding Quadrilaterals with Collinear Points

Any quadrilateral that includes 4 of the collinear \(m\) points will itself be degenerate. Therefore, we must exclude these cases: \(\binom{m}{4}\).

Step 9: Total Number of Different Quadrilaterals

The number of different quadrilaterals that can be formed is given by \(\binom{n}{4} - \binom{m}{4}\).

Key Concepts

Collinear Points

In geometry, collinear points refer to points that lie on the same straight line. Understanding collinearity is crucial when dealing with problems in plane geometry that involve determining possible lines or shapes, like triangles and quadrilaterals.

There are a few key characteristics of collinear points:

• If you can draw one straight line through two or more points, they are collinear.
• For some calculations, collinear points simplify the situation because they limit the number of forms or arrangements you must consider. For example, if three or more points are collinear, they cannot form a triangle.
• Identifying collinear points is critical when subtracting possible degenerate (not actually forming the intended shape) forms from total possible combinations.

If, in this exercise, we have 'm' collinear points, these points can only form one actual line. Other combinations of lines using these points need to be subtracted since they don't contribute additional new lines.

Combinations

Combinations are a fundamental concept in combinatorics, a branch of mathematics focused on counting, arranging, and analyzing possible configurations within a given set. They are used to determine how many ways you can choose a group of items from a larger collection, without regard to order.

In the formula for combinations, denoted as \(\binom{n}{k}\), the 'n' represents the total number of elements to choose from, and 'k' represents the number of elements to choose. The formula is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]This calculation allows one to compute:

• Lines that can be drawn by selecting any two out of 'n' points.
• Triangles by choosing any three points from 'n'.
• Quadrilaterals by selecting any four points from 'n'.

In exercises like the given problem, combinations help distinguish valid shapes from invalid ones due to collinearity.

Plane Geometry

Plane geometry is the study of shapes and figures in a flat, two-dimensional world. It mostly involves studying points, lines, angles, polygons (like triangles and quadrilaterals), and circles. Understanding how these geometric figures interact is crucial when solving problems related to them.

Some important features of plane geometry include:

• All points lie in a single plane, meaning any two points can always be connected by a straight line.
• It is possible to create various shapes by connecting points through lines.
• Analyzing properties of shapes, such as angles and side lengths, is typical for figuring out relationships and measurements.

Plane geometry builds the foundation for more advanced studies in geometry and is widely applicable, from architecture to engineering. In this exercise, understanding plane geometry is crucial not only for calculating the potential lines, triangles, and quadrilaterals but also for considering the impact of collinear points, which affect shape formation.