Problem 150 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(p\) and \(q\) are positive real numbers such that \(p^{2}+q^{2}\) \(=1\), then the maximum value of \((p+q)\) is \(\underline{\text { [2007 }}\) (A) 2 (B) \(1 / 2\) (C) \(\frac{1}{\sqrt{2}}\) (D) \(\sqrt{2}\)

Step-by-Step Solution

Answer

The maximum value of \(p+q\) is \(\sqrt{2}\) (option D).

Step 1: Analyze the Given Equation

We have the constraint equation \(p^2 + q^2 = 1\). This represents a circle of radius 1 centered at the origin in the \(pq\)-plane. We need to maximize the expression \(p + q\).

Step 2: Use Geometric Visualization

Since \(p^2 + q^2 = 1\) describes a circle, the line \(p+q = c\) (where \(c\) is a constant) will be a straight line with a slope of -1. We want to find the line \(p+q = c\) that is tangent to the circle, which occurs at the maximum \(c\).

Step 3: Determine Tangency Condition

The circle \(p^2 + q^2 = 1\) and the line \(p + q = c\) will be tangent to each other when the perpendicular distance from the origin to the line equals the radius of the circle (1 unit). The perpendicular distance from the origin to \(p + q = c\) is \(\frac{|c|}{\sqrt{2}}\).

Step 4: Equate Distance to Radius

Set the distance \(\frac{|c|}{\sqrt{2}} = 1\) (since the radius is 1). Solve for \(c\) to get \(c = \sqrt{2}\).

Step 5: Consider Positive Values

Since \(p\) and \(q\) must be positive, \(c\) remains positive as well. Therefore, the maximum possible value of \(p + q\) is \(\sqrt{2}\).

Key Concepts

Geometric Visualization

To solve the problem of finding the maximum value of \(p + q\) given the constraint \(p^2 + q^2 = 1\), we use a geometric approach. Think of the constraint as a circle with a radius of 1, centered at the origin of the \(pq\)-plane. The expression \(p + q = c\) forms a series of straight lines with a slope of -1. Our goal is to determine the line that is tangent to the circle, as tangency will indicate the maximum or minimum value.
This method helps visually appreciate how these elements interact:

• The circle \(p^2 + q^2 = 1\) represents all possible combinations of \(p\) and \(q\) that satisfy the equation.
• The line \(p + q = c\) provides a geometric method to adjust the values of \(p\) and \(q\).

The intersection points and resulting tangent line will dictate our maximum achievable sum of \(p + q\). This geometric visualization is a powerful tool that simplifies complex interactions among variables by representing them visually on a plane.

Tangency Condition

The next important concept in solving this problem is understanding the condition for tangency. For a line to be tangent to a circle, it must touch the circle at exactly one point. This property is key to maximizing \(p + q\). In our scenario:

• The circle is defined by \(p^2 + q^2 = 1\).
• The line is determined by the equation \(p + q = c\).
• Tangency implies that as the line moves outwards maintaining its slope, it will just touch the circle and not cross it.

The tangency condition is reached when the perpendicular distance from the origin (center of the circle) to our line \(p + q = c\) equals the radius of the circle. Verifying tangency is crucial to ensure we are maximizing the expression's value without exceeding the constraint.

Perpendicular Distance

The condition for tangency requires us to compute the perpendicular distance between a point (in this case, the origin) and a line. For line equation \(p + q = c\), its perpendicular distance to the origin can be calculated using the formula \(\frac{|c|}{\sqrt{2}}\). This distance is crucial because:

• It helps us evaluate when the line \(p + q = c\) just touches the circle.
• To satisfy the condition of tangency, we equate this distance to the circle's radius, which is 1.
• Solving for \(c\) gives \(c = \sqrt{2}\).

This calculation confirms that the maximum value of \(p + q\), constrained by the circle, is \(\sqrt{2}\). Understanding perpendicular distance in this context helps solidify the geometric reasoning behind the solution.