Question

If the focus of the parabola \[ (y - k)^2 = 4(x - h) \] always lies between the lines \(x + y = 1\) and \(x + y = 3\) then:

• A. \(0<h + k<2\)
• B. \(0<h + k<1\)
• C. \(1<h + k<2\)
• D. \(1<h + k<3\)

Hint:

Understanding how a parabola’s focus is derived from its equation is key to solving locus-related problems.

Answer 1

The Correct Option is A

The standard form of the given parabola is: \[ (y - k)^2 = 4(x - h) \] The focus of the parabola is given by: \[ (h + 1, k) \] Since the focus must lie between the lines \(x + y = 1\) and \(x + y = 3\), we substitute the focus into the inequalities: \[ 1<(h+1) + k<3 \] \[ 0<h + k<2 \] Thus, the required range for \( h + k \) is: \[ 0<h + k<2 \]

Answer 2

If the focus of the parabola
\[ (y - k)^2 = 4(x - h) \] always lies between the lines \(x + y = 1\) and \(x + y = 3\) then:
The equation of the parabola is of the form \((y - k)^2 = 4(x - h)\), which represents a parabola opening towards the right with focus at \((h + 1, k)\). We are given that the focus lies between the lines \(x + y = 1\) and \(x + y = 3\). To find the region where the focus lies, we substitute the coordinates of the focus into these line equations.
For the line \(x + y = 1\), the equation is: \[ x + y = 1 \quad \Rightarrow \quad h + 1 + k = 1 \quad \Rightarrow \quad h + k = 0 \] For the line \(x + y = 3\), the equation is: \[ x + y = 3 \quad \Rightarrow \quad h + 1 + k = 3 \quad \Rightarrow \quad h + k = 2 \] Thus, the condition that the focus lies between these two lines is: \[ 0 < h + k < 2 \] Therefore, the correct answer is:

\(0 < h + k < 2\)