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:
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:
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Understanding how a parabola’s focus is derived from its equation is key to solving locus-related problems.
Updated On: Jan 13, 2026
- \(0<h + k<2\)
- \(0<h + k<1\)
- \(1<h + k<2\)
- \(1<h + k<3\)
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The Correct Option is A
Approach Solution - 1
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:
\[
0<h + k<2
\]
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Approach Solution -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:
\[ (y - k)^2 = 4(x - h) \] 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:
\(0 < h + k < 2\)
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