Question

A rod having length \(2 \mathrm{c}\) moves along two perpendicular lines, then the locus of the midpoint of the rod is \(\ldots \ldots\) (a) \(x^{2}-y^{2}=c^{2}\) (b) \(x^{2}+y^{2}=c^{2}\) (c) \(x^{2}+y^{2}=2 c^{2}\) (d) None of these

Short answer

The locus of the midpoint of the rod is \(x^2 + y^2 = c^2\).

Step-by-step solution

Calculate the distance of M from the axes

First, we calculate the distance of midpoint M(x, y) from both the X and Y axes and denote these distances as a and b, respectively. The distances can be calculated using the coordinates: a = x b = y Now, consider the triangle OMA, where O is the origin (0, 0).

Apply Pythagorean theorem to OMA triangle

Since OMA is a right-angled triangle, we can use the Pythagorean theorem to relate distances a, b, and the half-length of the rod, c. The theorem states that the sum of the squares of the sides of a right-angled triangle equals the square of the hypotenuse. In this case, side OMx=c, side OA=a, and side AM=b. Using the Pythagorean theorem for triangle OMA, we get: a^2 + b^2 = c^2 Substitute x for a and y for b: x^2 + y^2 = c^2

Match the equation with given options

Now that we have the equation of the locus of M(x, y), compare it with the given options: \(x^2 + y^2 = c^2\) The equation matches option (b), therefore: The locus of the midpoint of the rod is \(x^2 + y^2 = c^2\).

Key concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly when dealing with right-angled triangles. At its core, the theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed in the simple equation:

\[ c^2 = a^2 + b^2 \]
where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the other two sides. In the context of the provided exercise, the Pythagorean theorem is used to understand the relationship between the coordinates of the midpoint of a moving rod and its distance from the axes in a coordinate plane.

• Identify the right-angled triangle within the problem.
• Assign the lengths of sides to the triangle's edges based on the given information.
• Apply the Pythagorean theorem to find the relationship between the sides.

This theorem is not just a theoretical principle but is frequently applied in real-world scenarios such as navigation, construction, and various fields of science and engineering. Understanding this rule is essential for solving many mathematical problems involving distances and measures.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This melds algebraic equations with geometric curves to make the calculation of distances and areas more systematic.

In coordinate geometry, points are described by their positions along the x (horizontal) and y (vertical) axes. The beauty of this system is that it allows us to solve geometric problems using algebra.

• Every point on the plane is associated with a pair of numerical coordinates (x, y).
• Geometric shapes can be represented through equations.

In the exercise we're discussing, the midpoint of the rod is such a point and its location is central to solving the problem. Here, the coordinates (x, y) not only tell us the midpoint's position but also aid in applying the Pythagorean theorem to find the locus of this moving midpoint.
Through coordinate geometry, one can determine shapes, sizes, and other properties of figures based on their equation. For instance, in this case, the equation \(x^2 + y^2 = c^2\) not only represents a set of points (the locus) but also defines a circle with radius \(c\) when graphed on a coordinate plane.

Right-Angled Triangles

Right-angled triangles are the simplest form of non-degenerate triangles where one of the angles measures exactly 90 degrees. The two sides that form the right angle are called the legs, and the side opposite the right angle is the hypotenuse, as in the Pythagorean theorem.

These triangles are crucial because they are the basis upon which trigonometry is built. Understanding right-angled triangles helps with grasping more complex geometric concepts and is indispensable in many technical fields.

• If you know the lengths of any two sides of a right-angled triangle, you can calculate the third.
• The Pythagorean theorem works exclusively with right-angled triangles.
• Trigonometric ratios such as sine, cosine, and tangent originate from right-angled triangles.

In our context, the right-angled triangle helps us derive the relationship between the length of the rod and its midpoint's coordinates. By seeing the movement of the rod along perpendicular lines through the lens of right-angled triangles, we are able to apply the Pythagorean theorem effectively and arrive at the midpoint's locus quite elegantly.