Problem 7 - A Complete Resource Book in Mathematics for JEE Main

Question

The centroid of the triangle formed by the feet of the normals from the point \((h, k)\) to the parabola \(y^{2}+4 a x\) \(=0,(a>0)\) lies on (A) \(x\)-axis (B) \(y\)-axis (C) \(x=h\) (B) \(y=k\)

Step-by-Step Solution

Answer

The centroid lies on the line \(x = h\).

Step 1: Understanding the Problem

We need to find the centroid of the triangle formed by the feet of the normals from a point \((h, k)\) to the parabola \(y^2 + 4ax = 0\) and determine where this centroid lies.

Step 2: Parameterize the Parabola

Express the parabola \(y^2 + 4ax = 0\) in parametric form as \((x, y) = (at^2, -2at)\), where \(t\) is a parameter.

Step 3: Equations of Normals from a Point

The equation of the normal to the parabola at \((at^2, -2at)\) is \(tx + y = 2at + atk^2\). Solve the system of equations consisting of the normal and point \((h, k)\) by substituting \(x = at^2\) and \(y = -2at\) into the normal's equation.

Step 4: Solving for Feet of the Normals

The coordinates where the normals intersect from\((h, k)\) to the parabola satisfy \(t(h-at^2) = k + 2at\). Using this equation, find the possible values of \(t\) and substitute them to determine the coordinates of the feet of the normals.

Step 5: Calculate Centroid Coordinates

With the coordinates of the feet of normals, say \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), calculate the centroid \((X, Y)\) using the formulas: \(X = \frac{x_1 + x_2 + x_3}{3}\) and \(Y = \frac{y_1 + y_2 + y_3}{3}\).

Step 6: Determine Location of Centroid

Analyze the expressions for \(X\) and \(Y\) derived from the previous step. Check whether \(X\) or \(Y\) equals any of the specified axis positions \(x = h\) or \(y = k\) or intersects either the \(x\)- or \(y\)-axis.

Key Concepts

Parabola

A parabola is a unique shape in coordinate geometry represented as the set of all points equidistant from a point called the focus and a line known as the directrix. The standard form of a parabola in the plane is generally given by the equation \[ y^2 = 4ax \] where 'a' is a constant that describes how wide or narrow the parabola is.

• In the given equation \(y^2 + 4ax = 0\), which translates to the equation \(y^2 = -4ax\), we recognize it as a horizontal parabola that opens to the left when \(a > 0\).

Understanding this geometric configuration helps in analyzing related problems, such as determining characteristics of lines—like normals—associated with it. Parabolas are pivotal in physics and engineering, capturing paths like the trajectory of projectile motion.

Normals to a Parabola

Normals are perpendicular lines to the tangent at any point on a curve. For a parabola, the normal line plays a crucial role in understanding the orientation and path of the curve. For a parabola defined as \(y^2 = 4ax\), at any point \((at^2, -2at)\), the normal line can be given by:

• \(tx + y = 2at + atk^2\)

This expression reflects how calculus not only allows us to find tangents (gradients) but also enables us to determine the perpendicular lines, which serve many applications in mathematical problems. In context, normals from a point to the parabola help form triangles whose centroid we're often interested in finding.
By solving the normals' equations simultaneously, we derive the intersection points, also known as the feet of the normals, which we use to form a triangle and calculate its centroid.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It combines algebra and geometry, making it a powerful tool for solving various problems in mathematics involving shapes and their properties. By employing the Cartesian plane, characterized by the 'x' and 'y' axes, we can graph and analyze equations of curves like parabolas efficiently.

• For instance, when exploring the properties of a parabola, or when solving for the centroid of a triangle in this context, coordinate geometry provides a structured approach.
• The centroid of a triangle, formed by points in \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), for example, can be easily calculated using the formula: \(X = \frac{x_1 + x_2 + x_3}{3}\) and \(Y = \frac{y_1 + y_2 + y_3}{3}\).

Understanding this blend of algebra and geometry allows for accurate prediction and description of geometric properties, important in fields ranging from computer graphics to architecture and beyond.