Question

A square of side a lies above the \(\mathrm{x}\) -axis and has one vertex at the origin. The side passing through the origin makes an angle a \(\alpha[0<\alpha<(\pi / 4)]\) with the positive direction of \(\mathrm{x}\) -axis. The eq. of its diagonal not passing through the origin is: (A) \(\mathrm{y}(\cos \alpha+\sin \alpha)+\mathrm{x}(\sin \alpha-\cos \alpha)=\mathrm{a}\) (B) \(\mathrm{y}(\cos \alpha+\sin \alpha)+\mathrm{x}(\sin \alpha+\cos \alpha)=\mathrm{a}\) (C) \(\mathrm{y}(\cos \alpha+\sin \alpha)+\mathrm{x}(\cos \alpha-\sin \alpha)=\mathrm{a}\) (D) \(y(\cos \alpha-\sin \alpha)-x(\sin \alpha-\cos \alpha)=a\)

Short answer

\(y(\cos\alpha + \sin\alpha) + x(\sin\alpha - \cos\alpha) = a\)

Step-by-step solution

Identify Points of the Square

Let's label the vertices of the square A, B, C, and D where A is at the origin (0, 0), B is the vertex connected to A with the side forming an angle α with the x-axis. From the position of vertex A, we can find the coordinates of vertices B and C (the opposite vertices of A and B). The coordinates for B can be calculated as: B(a*cos(α), a*sin(α)) The coordinates for C, the opposite vertex to A, on the square can be calculated as: C(a*cos(α + π/2), a*sin(α + π/2)) As we know, the angle between the sides of the square is π/2. So, we can utilize this property to find the coordinates of vertex C.

Find Equation of Diagonal

Now we have the coordinates of two opposite vertices of the square B and C. The line passes through these two points BC is the diagonal we are asked to find the equation of. Let's use the two-point form to find the equation: \(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)\) Let's plug in the coordinates of B (x1 = a*cos(α), y1 = a*sin(α)) and C (x2 = a*cos(α + π/2), y2 = a*sin(α + π/2)).

Simplify

Simplify the equation we obtained in the previous step by plugging in the coordinates, and using the following trigonometric identities: cos(α + π/2) = -sin(α) and sin(α + π/2) = cos(α). \(y - a\sin\alpha = \frac{a\cos\alpha - a\sin\alpha}{a(-\sin\alpha) - a\cos\alpha} (x - a\cos\alpha)\) After cancelling out a's and simplifying the equation, the final form is: \(y(\cos\alpha + \sin\alpha) + x(\sin\alpha - \cos\alpha) = a\) This equation matches option (A). So, the correct answer is: (A) \(y(\cos\alpha + \sin\alpha) + x(\sin\alpha - \cos\alpha) = a\)

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics where algebraic equations are used to describe and analyze geometric figures on a coordinate plane. In problems involving coordinate geometry, points are usually defined by their distance from two perpendicular axes, commonly denoted as the x-axis and y-axis.
In this exercise, we explored a square positioned on the coordinate plane. One vertex was at the origin (0,0) and another at point B, calculated using trigonometric references relative to the x-axis. The importance lies in leveraging these coordinates to address various properties of the square and lines formed by points.

• The square is defined by its vertices.
• The angle one side makes with the x-axis is a key element.
• Coordinates of each vertex can be calculated using trigonometric functions.

By understanding these basics, we can solve geometric problems by iteratively applying algebraic methods to find relationships, such as the equations of lines or diagonals.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions (like sine and cosine) that are true for all values of the involved variables. They are powerful tools in geometry, especially when you need to simplify expressions or find angles and lengths within geometric shapes.
For this problem, the identities played a crucial role in simplifying expressions when calculating the coordinates of the vertex C and the equation of the diagonal. Here are the relevant trigonometric identities:

• \[\cos(\alpha + \pi/2) = -\sin(\alpha)\]
• \[\sin(\alpha + \pi/2) = \cos(\alpha)\]

By substituting those trigonometric identities, we simplify complex expressions, a crucial step in forming correct and manageable geometric equations. Understanding and using these identities can significantly streamline processes when solving geometric equations.

Equations of Lines

Equations of lines are foundational aspects of coordinate geometry, used to describe relationships between geometric points on a plane. Typically, the equation of a line is expressed in the format \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
In this exercise, we focused on determining the equation of the square's diagonal, specifically passing through points B and C. Using the two-point form of a line's equation, we illustrate how we can derive line equations when given two points:\[y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)\]
This formula allows for direct calculation of a line by substituting in known coordinates, demonstrating how algebraic manipulation leads to discovering crucial geometric insights, such as finding that the preferred line equation in this solution strengthens our understanding of how lines can be expressed through given points.