Problem 50 - A Complete Resource Book in Mathematics for JEE Main

Question

In a circle of radius \(r\), an isosceles triangle \(A B C\) is inscribed with \(A B=A C\). If the \(\Delta A B C\) has perimeter \(P=\) \(2\left[\sqrt{2 h r-h^{2}}+\sqrt{2 h r}\right]\) and area \(A=h \sqrt{2 h r-h^{2}}\), \(57 .\) where \(h\) is the altitude from \(A\) to \(B C\), then \(\lim _{h \rightarrow 0^{-}} \frac{A}{P^{3}}\) is equal to (A) \(128 r\) (B) \(\frac{1}{128 r}\) (C) \(\frac{1}{64 r}\) (D) None of these

Step-by-Step Solution

Answer

(C) \(\frac{1}{64r}\)

Step 1: Understanding the Problem

We are given an isosceles triangle \(\triangle ABC\) inscribed in a circle of radius \(r\). The triangle has specific expressions for its perimeter \(P\) and area \(A\). We need to find \(\lim _{h \rightarrow 0^{-}} \frac{A}{P^{3}}\). Here, \(h\) is the altitude from \(A\) to \(BC\). This requires understanding how \(A\) and \(P\) behave as \(h\) approaches zero.

Step 2: Analyze Given Perimeter and Area

We have \(P = 2\left[\sqrt{2hr - h^2} + \sqrt{2hr}\right]\) and \(A = h \sqrt{2hr - h^2}\). These expressions indicate how \(P\) and \(A\) relate to \(h\). As \(h\) approaches 0, these expressions change, which will affect their ratio.

Step 3: Simplify Limit Calculation

To find \(\lim_{h \rightarrow 0^{-}} \frac{A}{P^{3}}\), analyze \(A\) and \(P\) for small \(h\). Substitute in expressions to simplify \(\frac{h \sqrt{2hr - h^2}}{\left(2[\sqrt{2hr - h^2} + \sqrt{2hr}]\right)^3}\). This requires focusing on the leading terms as \(h\) gets small.

Step 4: Evaluate Limit

Approximating as \(h \to 0^{-}\), the dominant terms are \(h \sqrt{2hr}\) for \(A\) and \(\sqrt{2hr}\) for \(\sqrt{2hr - h^2}\) in \(P\). Hence, \(\frac{A}{P^3} \approx \frac{h \sqrt{2hr}}{\left(4 \cdot \sqrt{2hr}\right)^3}\). Simplify this to \( \frac{h \sqrt{2hr}}{32hr \sqrt{2hr}}\), which further simplifies to \(\frac{1}{64rh}\). As \(h \to 0^{-}\), this limit simplifies to \(\frac{1}{64r}\), option (C).

Step 5: Conclude with Correct Answer

After simplification, we find \(\lim_{h \rightarrow 0^{-}} \frac{A}{P^3} = \frac{1}{64r}\). This corresponds to option (C).

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle that has two sides of equal length. In geometry, the points that make up this triangle are very important! Here’s how it works: the equal sides are typically labeled as legs, and the third side is known as the base. In triangle problems, like the one in the exercise, it's common to work with isosceles triangles because they often lead to simpler equations due to their symmetries.

In an isosceles triangle, the angles opposite the equal sides are also equal. This property is crucial when dealing with problems that involve calculating areas and perimeters, as these equal angles and sides can help simplify our solutions. When an isosceles triangle is inscribed in a circle (like in our exercise), a fascinating relationship emerges where the circle's center is aligned with the altitude dropping from the vertex angle to the base.

Key takeaways include:

• The equal sides help balance calculations.
• Symmetry is your friend in simplifying equations.
• Relationships with circumscribed circles can lead to interesting geometric insights.

Limits in Calculus

In calculus, limits help us understand the behavior of functions as they approach a certain point. The problem shared here deals with finding the limit as the altitude, represented by variable \(h\), approaches zero in our specific isosceles triangle problem.

The idea of a limit can be visualized by considering what happens to a function as its inputs get closer and closer to some value. It’s almost like tracing a curve with your finger towards a specific point on a graph! In our scenario, as \(h\) approaches zero, the expressions for the perimeter \(P\) and area \(A\) simplify dramatically, revealing the solution to the problem.

When working with limits:

• Pay attention to the behavior of expressions as variables approach a boundary value.
• Focus on the dominant terms when a variable approaches infinity or zero that will drive the limit.
• This skill is essential for solving many real-world problems where predicting trends and behaviors is crucial.

Perimeter and Area of Triangle

Calculating the perimeter and area of a triangle helps us determine its size and space it covers, which are key for countless applications in geometry. In the problem at hand, formulas for both perimeter \(P\) and area \(A\) of the triangle \(\Delta ABC\) are given, depending on the altitude \(h\) and the radius of the circumscribing circle \(r\).

For perimeter, the expression is \(P = 2[\sqrt{2hr - h^2} + \sqrt{2hr}]\), and for area, \(A = h \sqrt{2hr - h^2}\). These might look complex at first, but they actually simplify nicely when analyzed as the triangle flattens (leading \(h\) towards zero).

Here are some additional insights:

• Perimeter measures the boundary length, requiring all sides' lengths.
• Area measures the enclosed space, useful in understanding proportions and densities.
• Simplifying such expressions can often involve recognizing underlying geometric patterns.