Problem 10 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{S}\) be the set of all triangles in the \(\mathrm{xy}\)-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in \(\mathrm{S}\) has area \(50 \mathrm{sq}\). units, then the number of elements in the set \(\mathrm{S}\) is: (a) 9 (b) 18 (c) 36 (d) 32

Step-by-Step Solution

Answer

The number of elements in set \(S\) is 9.

Step 1: Understand Triangle Area Formula

For a triangle with vertices at \((0,0)\), \((a,0)\), and \((0,b)\), the area can be calculated as \( \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times a \times b \).

Step 2: Set the Area Equation

Given that each triangle has an area of 50 square units, we set up the equation based on the area formula: \( \frac{1}{2} \times a \times b = 50 \). Simplifying this, we get \( a \times b = 100 \).

Step 3: Find Integral Solutions for a and b

We need to find all pairs \((a, b)\) where both \(a\) and \(b\) are positive integers such that \(a \times b = 100\). The pairs are: \((1, 100), (2, 50), (4, 25), (5, 20), \) \((10, 10), (20, 5), (25, 4), (50, 2), (100, 1)\).

Step 4: Determine Symmetrical Pairs

Each pair \((a, b)\) can be interchanged to form another valid triangle, giving symmetrically opposite pairs such as \((1, 100)\) with \((100, 1)\), etc., except \((10, 10)\), which remains the same since it is symmetrical in itself.

Step 5: Count the Total Pairs

By listing all unique ordered pairs and their symmetric counterparts, we have: \((1, 100), (100, 1), (2, 50), (50, 2), (4, 25), \) \((25, 4), (5, 20), (20, 5), (10, 10)\). Total pairs = 9.

Key Concepts

Integral Coordinates

Integral coordinates refer to coordinates with integer values for each point on a grid. In the context of this exercise, we're dealing with triangles that have vertices at the origin \((0,0)\) and on the coordinate axes. This means that the coordinates of these vertices are integral, such as \((a,0)\) or \((0,b)\), where \(a\) and \(b\) are positive integers.
These integral points make calculations straightforward, especially when dealing with a set condition like in this exercise, where the area of the triangles is fixed. This simplifies finding possible integer solutions because we only deal with whole numbers.
In practical problems such as this, you may be required to find all possible integer coordinate points that satisfy a given equation or condition, like the fixed area of 50 square units here. Understanding that some solutions can swap their x and y values due to symmetry can also multiply the number of valid solutions.

Triangle Area

The area of a triangle in coordinate geometry can be calculated using various methods. However, when the vertices of the triangle are conveniently placed on the coordinate axes, the area formula simplifies significantly.
The formula for the area when one vertex is \((0,0)\) and the other vertices are \((a,0)\) and \((0,b)\) is:

• Area = \( \frac{1}{2} \times a \times b \)

This formula is easy to grasp because it turns the problem into a simple base-time-height calculation. The base and height correspond to the lengths of the legs lying on the x-axis and y-axis, respectively.
When you know the area (50 square units in this exercise), you can set this equation to find compatible \(a\) and \(b\) values that satisfy \(\frac{1}{2} \times a \times b = 50\). Essentially, this gives us \(a \times b = 100\), which is the key relation to identify potential integral pairs that satisfy this equation.

Symmetrical Pairs

In this exercise, determining symmetrical pairs is crucial in counting unique triangles. Symmetrical pairs occur when you have two vertices swapped on the coordinate axes.
For example, if you have a vertex at \((a, 0)\) and another at \((0, b)\), swapping these points to \((0, a)\) and \((b, 0)\) gives you a symmetrical pair. Both configurations are valid triangles since the area remains unchanged and equals 50 square units.
Identifying symmetrical pairs is essential because it effectively doubles your solution set from the initially calculated pairs, except for cases where \(a = b\), like \((10, 10)\) here, which is its own symmetrical version.
When listing symmetrical pairs, ensure each pair is counted only once in either order unless it's symmetrical to itself. This understanding leads to an accurate count of triangles, as shown by extending pairs such as \((1, 100)\) and \((100, 1)\), ensuring none are missed, thereby arriving at the correct answer of 9 triangles.