Problem 13 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{ABC}\) be a triangle whose circumcentre is at \(\mathrm{P}\). If the position vectors \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{P}\) are \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\) and \(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{4}\) respectively, then the position vector of the orthocentre of this triangle, is: \(\quad\) Online April 10, 2016] (a) \(-\left(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{2}\right)\) (b) \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}\) (c) \(\left(\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}}{2}\right)\) (d) \(\overrightarrow{0}\)

Step-by-Step Solution

Answer

None of the provided options match exactly, though approximations or errors should be checked in the context.

Step 1: Understand Given Vectors

We are provided with the position vectors of points A, B, C, and P, which represents the circumcentre of the triangle ABC. According to the problem, \( \overrightarrow{P} = \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{4} \).

Step 2: Recall Properties of the Circumcentre

In a triangle, the position vector of the circumcentre \( \overrightarrow{P} \) can be expressed using the average formula of the three vertices if the triangle is equilateral. In non-equilateral triangles, \( P \) is at a varying ratio closer to the side opposite the largest angle.

Step 3: Determine the Orthocentre Expression

The position vector of the orthocentre \( \overrightarrow{H} \) for any triangle can be derived by using the fact that \( \overrightarrow{H} + \overrightarrow{P} = \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} \). Thus, \( \overrightarrow{H} = (\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}) - \overrightarrow{P} \).

Step 4: Substitute Circumcentre Vector

Use the provided position vector of the circumcentre, \( \overrightarrow{P} = \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{4} \), in the equation for the orthocentre. This gives: \( \overrightarrow{H} = \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} - \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{4} \).

Step 5: Simplify the Expression

Calculate \( \overrightarrow{H} \) by simplifying the equation: \[ \overrightarrow{H} = \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} - \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{4} = \frac{4(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c})}{4} - \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{4} = \frac{3(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c})}{4} \].

Step 6: Consider Options

Evaluate the calculated orthocentre vector expression \( \frac{3(\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c})}{4} \) against the provided options. This does not directly match any of the given choices, indicating the orthocentre derived requires interpretation or an approximation to available choices.

Key Concepts

Position Vectors

Position vectors are a way to describe the location of a point relative to an origin. They are usually denoted by an arrow symbol (e.g., \( \overrightarrow{a} \), \( \overrightarrow{b} \), \( \overrightarrow{c} \)). These vectors point from the origin to the respective points A, B, and C, which form the vertices of our triangle. This makes understanding and handling orientations straightforward in a triangle.

Position vectors help us to easily perform algebraic operations on points in a triangle, such as addition or subtraction. For instance, if you know the position vectors of two points, say A and B, you can find the vector pointing from A to B by computing \( \overrightarrow{b} - \overrightarrow{a} \). This capability is crucial for calculating other key points like the orthocentre or circumcentre.

• Each vertex of the triangle ABC has its unique position vector.
• These vectors can be used to find other important points using algebraic operations.

Circumcentre

The circumcentre is the point that is equidistant from all three vertices of a triangle. This point is the center of the circle in which the triangle can be inscribed, known as the circumcircle.

In vector terms, finding the circumcentre is often simplified when the triangle is equilateral, in which case, the circumcentre can be found by averaging the position vectors: \[ \overrightarrow{P} = \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{3} \]. However, for general triangles, the circumcentre calculation might need some adjustments depending on triangles' angles and side lengths.

• It is equidistant from all vertices of the triangle.
• If the triangle is equilateral, circumcentre is at the centroid.
• The position vector of the circumcentre, \( \overrightarrow{P} \), is given in the problem as \( \frac{\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}}{4} \).

Triangle Properties

Triangles have distinct properties that are useful when solving problems related to them, especially in vector algebra. Here are a few relevant properties:

• The sum of the angles in any triangle is always \(180^\circ\).
• The centroid, orthocentre, and circumcentre are key points related to triangles, each serving a unique function.
• The orthocentre is the point where the altitudes intersect, while the centroid is the point where medians intersect. Both lie along the Euler line with the circumcentre.

Understanding these properties aids in making sense of relationships like the expression \( \overrightarrow{H} + \overrightarrow{P} = \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} \), which relates the centroid and orthocentre and is specific to certain triangle layouts.

Vector Algebra

Vector algebra is a branch of mathematics that deals with quantities that have both magnitude and direction. In a triangle context, it helps to describe positions and evaluate relationships between points and lines.

Vector operations such as addition, subtraction, and multiplication by scalars enable us to manipulate position vectors in meaningful ways. For example, we determined the orthocentre's position by manipulating the vectors of the triangle's vertices and the circumcentre as seen in the solution.
Bullets summarizing the key operations in vector algebra:

• Addition: Combining two vectors to determine a resultant vector.
• Subtraction: Finding the difference between two vectors to determine a directional vector.
• Scalar Multiplication: Resizing a vector by multiplying it with a scalar value, which affects its magnitude while maintaining its direction.

Understanding these operations helps in analyzing and solving problems related to position vectors and the geometric entities of a triangle like the orthocentre and circumcentre.