Problem 88 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Two vertices of an equilateral triangle are \((-1,0)\) and (1,0). An equation of its circumcentre is (A) \(x^{2}+y^{2}+\frac{2}{\sqrt{3}} y-1=0\) (B) \(x^{2}+y^{2}-\frac{2}{\sqrt{3}} y-1=0\) (C) \(x^{2}+y^{2}+\frac{2}{\sqrt{3}} y+1=0\) (D) none of these

Step-by-Step Solution

Answer

The correct equation for the circumcentre is (B) \(x^{2}+y^{2}-\frac{2}{\sqrt{3}} y-1=0\).

Step 1: Identify Midpoint of Base

The base of the equilateral triangle is the line segment between the points \((-1,0)\) and \((1,0)\). To find the midpoint, use the midpoint formula: \(\left(\frac{-1+1}{2}, \frac{0+0}{2}\right) = (0,0)\).

Step 2: Calculate Length of Base

The distance between the vertices \((-1,0)\) and \((1,0)\) is calculated with the distance formula: \(\sqrt{(1 + 1)^2 + (0 - 0)^2} = \sqrt{4} = 2\).

Step 3: Determine Height of the Triangle

For an equilateral triangle, the height can be calculated using the formula for the height \(h = \frac{\sqrt{3}}{2} a\), where \(a\) is the side's length. Here, the height \(h = \frac{\sqrt{3}}{2} \times 2 = \sqrt{3}\).

Step 4: Find Circumcentre Coordinates

The circumcentre is located at the centroid in an equilateral triangle, coinciding with the height's intersection from the vertex opposite the midpoint. From the midpoint (0,0), move \(\sqrt{3}\) units in the vertical direction, giving the circumcentre coordinates as \((0, \sqrt{3})\) or \((0, -\sqrt{3})\).

Step 5: Choose Equation for Circumcentre Through Origin

The circumcentre lies on the perpendicular bisector of the base, equidistant from both base vertices. Considering the symmetry along x-axis, let it be \((0, \sqrt{3})\). Substituting different options in x=0, check compatibility:\((0)^2 + (\sqrt{3})^2 + \frac{2}{\sqrt{3}} \times \sqrt{3} - 1 = 3 + 2 - 1 = 4\ (false)\),\((0)^2 + (\sqrt{3})^2 - \frac{2}{\sqrt{3}} \times \sqrt{3} - 1 = 3 - 2 - 1 = 0\ (true)\).

Key Concepts

Circumcentre

An equilateral triangle, such as the one described with vertices i(-1,0) and (1,0), has a unique point called the circumcentre. The circumcentre is the point where the perpendicular bisectors of the sides intersect. In an equilateral triangle, due to its symmetry, the circumcentre coincides with both the centroid and orthocenter. For this particular triangle, after finding the midpoint and determining the height dimension, the circumcentre can be positioned at either (0, \(\sqrt{3}\)) or (0, -\(\sqrt{3}\)). The circumcentre’s role is vital in understanding the balance and symmetry of the triangle, ensuring that all vertices are equidistant from this central point.

Midpoint Formula

The midpoint formula is an essential mathematical tool used to find the center point, or midpoint, of a line segment. This midpoint is particularly important in geometry when working with shapes like triangles. The formula is given by:\[\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]For our equilateral triangle exercise, the base is defined by the vertices (-1,0) and (1,0). To find the midpoint of this base, we apply the formula:\[\left(\frac{-1+1}{2}, \frac{0+0}{2}\right) = (0,0)\]This result shows that the midpoint of the base is exactly at the origin (0,0). The midpoint is foundational to finding other elements of the triangle, such as its height and circumcentre.

Distance Formula

The distance formula is another cornerstone of geometric calculations. It is used to compute the length of a line segment given its endpoints. The formula is:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In the case of our triangle, the base segment connects (-1,0) and (1,0). Substituting these coordinates into the distance formula, we find:\[\sqrt{(1 - (-1))^2 + (0 - 0)^2} = \sqrt{(2)^2} = \sqrt{4} = 2\]This calculation establishes that the length of the base side is 2 units. Knowing the side length is crucial when calculating the height of the triangle and its circumcentre.

Height of Triangle

The height of an equilateral triangle can be determined using a specific formula derived from its properties. The formula is:\[h = \frac{\sqrt{3}}{2} a\]where \(a\) is the side length of the triangle. For our equilateral triangle, with a side length \(a = 2\), the height is calculated as follows:\[h = \frac{\sqrt{3}}{2} \times 2 = \sqrt{3}\]This formula reflects the intrinsic triangular symmetry and ensures correct perpendicular distance from the base to the opposite vertex. The height plays a critical role in finding the coordinate of the circumcentre by determining how far vertically we should position the circumcentre in a coordinate plane.