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: Mar 26, 2025
- \(0<h + k<2\)
- \(0<h + k<1\)
- \(1<h + k<2\)
- \(1<h + k<3\)
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The Correct Option is A
Solution and Explanation
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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