Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the focus of the parabola \((y-\beta)^{2}=4(x-\alpha)\) always lies between the lines \(x+y=1\) and \(x+y=3\), then (A) \(1<\alpha+\beta<2\) (B) \(0<\alpha+\beta<1\) (C) \(0<\alpha+\beta<2\) (D) none of these

Step-by-Step Solution

Answer

(C) \(0<\alpha+\beta<2\)

Step 1: Identify Elements of the Parabola

The given equation of the parabola is \((y-\beta)^{2}=4(x-\alpha)\). This is a horizontal parabola with vertex \((\alpha, \beta)\). The focus of a standard horizontal parabola \((y-k)^2 = 4p(x-h)\) is \((h+p, k)\). For our parabola, \(h=\alpha\), \(k=\beta\), and \(p=1\), so the focus is at \((\alpha+1, \beta)\).

Step 2: Analyze the Position of the Focus Lines

The focus \((\alpha+1, \beta)\) must lie between the lines \(x+y=1\) and \(x+y=3\). This means the \(y\)-coordinate \(\beta\) must satisfy both \((\alpha+1) + \beta = 1\) and \((\alpha+1) + \beta = 3\). Therefore, \((\alpha+1) + \beta\) must be between 1 and 3.

Step 3: Solve the Inequality for \(\alpha+\beta\)

Re-arranging the inequality \(1 < (\alpha+1) + \beta < 3\) results in \(1 < \alpha + 1 + \beta < 3\). By simplifying, we have \(0 < \alpha + \beta < 2\).

Step 4: Determine the Correct Answer

Review the options given: (A)\(1<\alpha+\beta<2\), (B)\(0<\alpha+\beta<1\), (C)\(0<\alpha+\beta<2\). From the derived inequality, the valid condition is \(0 < \alpha + \beta < 2\), which matches option (C).

Key Concepts

Focus of parabola

Parabolas are fascinating geometric shapes that have a unique point known as the focus. The focus plays a crucial role in defining the properties of a parabola. For a standard equation of a horizontal parabola \[(y-k)^2 = 4p(x-h)\]we find the focus using the formula \[(h+p, k)\].Understanding the focus is vital. It is the point from which distances to any point on the parabola are measured. This creates the reflective property of a parabola, where paths from the focus reflect off the curve and travel parallel to the axis of symmetry. In the context of our exercise, the focus \((\alpha+1, \beta)\)must lie between two specific lines. This aspect of focus positioning is crucial for solving the problem and understanding the relationship between the parabola and external constraints. In more practical terms, understanding the focus helps in designing parabolic mirrors and antennas, which use the reflective properties of parabolas to direct signals and light.

Vertex of parabola

The vertex of a parabola is like its heart. It is the point where the curve changes direction, meaning it is either the highest or lowest point (depending on the orientation) of the parabola. For a parabola described by the equation \[(y-k)^2 = 4p(x-h)\]the vertex is simply \((h, k)\).In our specific exercise, the vertex is found at \((\alpha, \beta)\).This vertex is significant since it acts as a central reference point to locate the focus and sketch the parabola's shape. By understanding the position of the vertex, one can easily graph the parabola and analyze how it interacts with other geometric elements, such as the lines mentioned in our problem.Vertices are also relevant in optimization problems, where identifying the highest or lowest points can be crucial, such as in profit maximization or minimizing distances in physics and engineering.

Inequalities in geometry

Inequalities are often used in geometry to describe relative positioning and relationships between different elements. They help specify the bounds within which a certain geometric entity lies. For example, in our exercise, we use inequalities to understand where the focus of the parabola is positioned concerning two lines.The inequality \[1 < (\alpha + 1) + \beta < 3\]translates into a condition for \((\alpha + \beta)\)to satisfy specific positional requirements. By solving this, we find that\[0 < \alpha + \beta < 2\]which indicates that the sums of the coordinates must lie within these bounds.This kind of problem-solving approach where inequalities delimit regions is common in optimization and analysis within geometry. Using inequalities, we can determine feasibility and constraints, playing a fundamental role in fields such as computer graphics, architectural design, and any situation where a specific geometric configuration must be preserved.