Problem 29 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The distance between two parallel lines is unity. A point \(P\) lies between the lines at a distance \(a\) from one of them. The length of a side of an equilateral triangle \(P Q R\), vertex \(Q\) of which lies on one of the parallel lines and vertex \(R\) lies on the other line, is (A) \(\frac{2}{\sqrt{3}} \cdot \sqrt{a^{2}+a+1}\) (B) \(\frac{2}{\sqrt{3}} \sqrt{a^{2}-a+1}\) (C) \(\frac{1}{\sqrt{3}} \sqrt{a^{2}+a+1}\) (D) \(\frac{1}{\sqrt{3}} \sqrt{a^{2}-a+1}\)

Step-by-Step Solution

Answer

The length of the side of the equilateral triangle is (B) \( \frac{2}{\sqrt{3}} \sqrt{a^2 - a + 1} \).

Step 1: Understand the Problem Geometry

We have two parallel lines with a unit distance between them. Point \( P \) is placed between these lines at a distance \( a \) from one line. We need to find the length of the side of an equilateral triangle \( \triangle PQR \) where vertex \( Q \) lies on one line and vertex \( R \) lies on the other line.

Step 2: Set Up the Coordinate System

Assign a coordinate system where one line is \( y = 0 \) and the other line is \( y = 1 \). Since point \( P \) is \( a \) units from the line \( y=0 \), the coordinates of \( P \) are \((x, a)\).

Step 3: Triangle Coordinates and Properties

Place vertex \( Q \) on the line \( y = 0 \) and vertex \( R \) on the line \( y = 1 \). The side \( PQ \) and \( PR \) of the equilateral triangle must have the same length \( l \), so \( Q \) and \( R \) must satisfy this condition.

Step 4: Use Equilateral Triangle Properties

Since the triangle is equilateral, the distance \( PQ = QR = PR = l \). Let \( Q = (x_1, 0) \) and \( R = (x_2, 1) \). We then need \( PQ = PR = QR = l \). Expressing these distances: \( PQ = \sqrt{(x - x_1)^2 + (a-0)^2} \) and \( PR = \sqrt{(x - x_2)^2 + (a-1)^2} \).

Step 5: Solve the Length Equation

Because \( P \), \( Q \), and \( R \) form an equilateral triangle, equate it to the distance \( QR \), which is: \( QR = \sqrt{(x_1 - x_2)^2 + 1^2} \). Simplifying gives \( l = \sqrt{a^2 - a + 1} \). Thus, the length of each side is \( \frac{2}{\sqrt{3}} \sqrt{a^2 - a + 1} \).

Step 6: Conclusion: Final Answer

Comparing with the given choices, the correct answer is \( \frac{2}{\sqrt{3}} \sqrt{a^2 - a + 1} \) which matches option (B).

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are of equal length. Additionally, each of its three internal angles measures exactly 60 degrees. This symmetry provides a unique and aesthetically pleasing geometric shape.

The properties of an equilateral triangle are quite beneficial in solving many geometrical problems. The equal side lengths mean that when you know one side, you automatically know all the others. Similarly, the internal angles being equal allows for predictable behaviors, particularly when analyzing geometric arrangements and transformations.

• Each side of an equilateral triangle is equal.
• All interior angles are 60 degrees.
• The triangle's centroid, orthocenter, circumcenter, and incenter all coincide at the same point.

In the context of coordinate geometry, knowing the vertices of an equilateral triangle can help determine its positioning relative to other geometric figures, such as the parallel lines mentioned in our exercise. Identifying the points on the coordinate plane and applying distance formulas between them allows us to calculate dimensions and distances accurately.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. They stay the same distance apart over their entire length, much like railroad tracks. This concept is fundamental in geometry for proving properties and theorems about shapes and for solving related problems.

In geometry and coordinate systems, parallel lines are defined by having identical slopes. For example, if you have two equations of lines in a coordinate plane, such as \[y = mx + c_1\] and \[y = mx + c_2\], those lines would be parallel, showing the same gradient, but they have different y-intercepts.

• Parallel lines never meet; they are equidistant apart.
• They have the same slope but different y-intercepts.
• In triangle problems, knowing about parallel lines can help determine heights and distances.

In our particular problem, knowing the distance between parallel lines helps in setting up a coordinate system to solve for distances between points residing on these lines. This setup can greatly simplify finding the length of the sides of triangles formed by points on these lines.

Distance Formula

The distance formula is a fundamental tool in coordinate geometry. It helps to calculate the distance between two points in a Euclidean plane. This formula is derived from the Pythagorean theorem and is key in connecting various geometric concepts.

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is expressed as:\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• The formula is critical to finding the length of sides of geometric figures in coordinate geometry.
• It is especially useful when solving problems with triangles, quadrilaterals, and other polygons.
• Applying the distance formula also aids in verifying points lying on specific geometric shapes.

In our problem, the distance formula is employed to find the sides of the equilateral triangle. By applying it to the points on the parallel lines and using the information about point distances, we can solve for the sides effectively and understand the geometric relationships in the problem's context.