Problem 76 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(g(x)\) be a function defined on \([-1,1]\). If the area of the equilateral triangle with two of its vertices at \((0,0)\) and \((x, g(x))\) is \(\frac{\sqrt{3}}{4}\), then the function is (A) \(g(x)=\pm \sqrt{\left(1-x^{2}\right)}\) (B) \(g(x)=-\sqrt{\left(1-x^{2}\right)}\) (C) \(g(x)=\sqrt{\left(1-x^{2}\right)}\) (D) \(g(x)=\sqrt{\left(1+x^{2}\right)}\)

Step-by-Step Solution

Answer

The function is \(g(x) = \pm \sqrt{1 - x^2}\), corresponding to option (A).

Step 1: Understand the Problem

We are given a function \(g(x)\) and we need to find it by examining the condition that two vertices of an equilateral triangle are at \((0,0)\) and \((x, g(x))\). This triangle has a specified area of \(\frac{\sqrt{3}}{4}\).

Step 2: Recall Area Formula for Equilateral Triangle

For an equilateral triangle with side length \(a\), the area \(A\) is given by \(A = \frac{\sqrt{3}}{4}a^2\). Since the given area is \(\frac{\sqrt{3}}{4}\), we have the equation \(\frac{\sqrt{3}}{4}a^2 = \frac{\sqrt{3}}{4}\).

Step 3: Solve for Side Length \(a\)

Equating the area formula to the given area, \(a^2 = 1\), hence \(a = 1\). Therefore, \((x, g(x))\) must be \(1\) unit in distance from \((0,0)\).

Step 4: Use Distance Formula

The distance between two points \((0,0)\) and \((x, g(x))\) is \(\sqrt{x^2 + g(x)^2} = 1\). Thus, we have the equation \(x^2 + g(x)^2 = 1\).

Step 5: Solve for \(g(x)\)

Rearranging \(x^2 + g(x)^2 = 1\), we find \(g(x)^2 = 1 - x^2\). Therefore, \(g(x) = \pm \sqrt{1 - x^2}\).

Step 6: Choose the Correct Option

The solution \(g(x) = \pm \sqrt{1 - x^2}\) corresponds to option \(A\). Therefore, the correct function is \(g(x) = \pm \sqrt{1 - x^2}\).

Key Concepts

Equilateral Triangle Area

The area of an equilateral triangle is a fascinating topic in geometry. An equilateral triangle has all three sides of equal length, and this symmetry gives it unique properties. The formula for finding the area of an equilateral triangle with side length \(a\) is \(\frac{\sqrt{3}}{4}a^2\). This formula arises from splitting the triangle into two 30-60-90 triangles, where the height can be calculated using half of the base and the Pythagorean theorem.

• The side length \(a\) acts as both the base and the height of the equilateral triangle due to symmetry.
• The height \(h\) is found by \(h = \frac{\sqrt{3}}{2}a\).
• Substituting \(h\) into the standard area formula \(\frac{1}{2} \times \text{base} \times \text{height}\) results in the special formula for an equilateral triangle.

This specific exercise provides the area \(\frac{\sqrt{3}}{4}\), indicating a side length of \(1\), confirming its vertices are as detailed in the next steps.

Distance Formula

The distance formula is a crucial tool for determining the distance between two points in a coordinate plane. It's derived from the Pythagorean theorem and given by \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). In this problem, we use it to find the distance between the origin \((0,0)\) and point \((x, g(x))\).

• Substituting the coordinates, we calculate the distance as \( \sqrt{x^2 + g(x)^2} \).
• Given that an equilateral triangle requires equal sides, this distance is equal to \(1\), per the information derived from the area.

Hence, we derive the condition \( x^2 + g(x)^2 = 1 \), which will lead us to uncovering the function \(g(x)\).

Function Analysis

Function analysis allows us to explore the properties and behavior of a mathematical function. In this exercise, we determine the nature of the function \(g(x)\). Based on the condition derived from the distance formula:\[ x^2 + g(x)^2 = 1 \]

• This is the equation of a circle centered at the origin with a radius of \(1\).
• Rearranging gives \( g(x)^2 = 1 - x^2 \).
• Taking the square root results in \( g(x) = \pm \sqrt{1 - x^2} \).

Option \(A\) correctly states that \(g(x)\) can be both \( + \sqrt{1 - x^2} \) and \( - \sqrt{1 - x^2} \), representing the top and bottom halves of the circle respectively. Understanding and solving these steps gives insight into how different vertices of a geometric figure can impose constraints on a function's form.