Problem 114 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Three forces \(\vec{P}, \bar{Q}\) and \(\vec{R}\) acting along \(L A, I B\) and \(I C\), where \(I\) is the incentre of a \(\triangle A B C\) are in equilibrium. Then \(\vec{P}: \vec{Q}: \vec{R}\) is (a) \(\operatorname{cosec} \frac{A}{2}: \operatorname{cosec} \frac{B}{2}: \operatorname{cosec} \frac{C}{2}\) (b) \(\sin \frac{A}{2}: \sin \frac{B}{2}: \sin \frac{C}{2}\) (c) \(\sec \frac{A}{2}: \sec \frac{B}{2}: \sec \frac{C}{2}\) (d) \(\cos \frac{A}{2}: \cos \frac{B}{2} ; \cos \frac{C}{2}\)

Step-by-Step Solution

Answer

(a) \(\operatorname{cosec} \frac{A}{2}: \operatorname{cosec} \frac{B}{2}: \operatorname{cosec} \frac{C}{2}\)

Step 1: Understanding the Concept of Equilibrium

When three forces are in equilibrium, the vector sum of these forces equals zero. Here, the forces \(\vec{P}\), \(\vec{Q}\), and \(\vec{R}\) act along the lines connecting the incenter \(I\) to the vertices \(A\), \(B\), and \(C\) of triangle \(\triangle ABC\).

Step 2: Relating Forces to Sides of the Triangle

The forces \(\vec{P}, \vec{Q},\) and \(\vec{R}\) are proportional to the sides opposite the angles at \(A\), \(B\), and \(C\) respectively. Due to the properties of the incenter, the magnitude of a force is inversely proportional to the sine of half the angle opposite to the side.

Step 3: Applying the Incenter Theorem

Since \(I\) is the incenter, the distances from \(I\) to the sides are proportional to the sines of the half-angles of \(\triangle ABC\). Therefore, the magnitudes of \(\vec{P}, \vec{Q},\) and \(\vec{R}\) are proportional to \(\csc \frac{A}{2}\), \(\csc \frac{B}{2}\), and \(\csc \frac{C}{2}\) respectively.

Step 4: Determining the Correct Proportional Relationship

Given the relationships derived from the equilibrium condition and properties of the incenter, the proportional relationship between the forces should be expressed in terms of the cosecant of the half-angles: \(\vec{P}:\vec{Q}:\vec{R} = \operatorname{cosec} \frac{A}{2}: \operatorname{cosec} \frac{B}{2}: \operatorname{cosec} \frac{C}{2}\).

Key Concepts

Equilibrium

When we talk about equilibrium in the context of forces acting on a point, such as the incenter of a triangle, it means that the forces balance each other out perfectly. In a more technical sense, equilibrium occurs when the vector sum of all forces acting at that point equals zero. This can happen when the forces have both the same magnitude and exact opposite directions, or when multiple forces interact in such a way that their combined effect cancels to zero.

In the case of the problem at hand, three forces (\(\vec{P}, \vec{Q}, \vec{R}\)) act upon the incenter of triangle \(\triangle ABC\). These forces, when in equilibrium, don't cause any motion because their effects collectively balance out to zero. This underpins many concepts in physics and engineering where stability is crucial, such as bridges, buildings, and even ships staying upright in water.

• Equilibrium requires careful calculation to ensure that all forces perfectly counteract each other.
• The study of equilibrium helps us design structures and systems that remain steady and safe under varying conditions.

Incenter of a Triangle

The incenter of a triangle is a very special point. It is defined as the point where the angle bisectors of the triangle's interior angles meet. The incenter is at an equal distance from all the sides of the triangle and can be thought of as the center of the circle that is inscribed in the triangle (known as the incircle).

Understanding the incenter's characteristics helps solve numerous geometry problems. For instance, if we know where the incenter is, we can determine the radius of the incircle with the formula: radius = area divided by the semi-perimeter. In our context of force equilibrium, the incenter's relevance is tied to the relationship between the forces \(\vec{P}, \vec{Q}, \vec{R}\), and the triangle's angles.

• The incenter divides any angle into two equal parts, providing symmetry and balance.
• This point is always inside the triangle, irrespective of the triangle's type (acute, right, or obtuse).

Cosecant

Cosecant is a trigonometric function, often abbreviated as \(\csc\), and it is the reciprocal of the sine function. Thus, for any given angle \(\theta\), \(\csc(\theta) = 1/\sin(\theta)\).

The cosecant function, like all trig functions, is crucial in various calculations involving angles and distances, particularly in right-angled triangles. In our triangle situation, cosecant relates to the forces' magnitudes at equilibrium. Specifically, the magnitudes of forces \(\vec{P}, \vec{Q},\) and \(\vec{R}\) are expressed in terms of the cosecant of half the angles \(A\), \(B\), and \(C\) respectively.

• The cosecant function is undefined for angles where the sine is zero, that is at multiples of \(180^\circ\).
• Understanding \(\csc\) can help solve problems revolving around periodic properties and wave-like patterns.

Sine of Half-Angle

The sine of a half-angle is a concept that stems from the half-angle identities in trigonometry. It's particularly used when dealing with problems involving circle properties, such as circular sectors or inscribed circles.

In our exercise, the sine of half-angles directly influences the equilibrium condition of forces because the forces connecting the incenter to triangle vertices are determined based on these angles. The importance arises from the property that the magnitudes of the forces are inversely proportional to the sine of the half-angles. This is why the ratios for the forces are given in terms of the cosecant (or reciprocal of sine) of these half-angles.

• The half-angle identities are useful for simplifying and solving complex trigonometric expressions.
• Using half-angles gives a straightforward approach to find exact values of various functions for specific angles.