Question:
Which of the following represents a parabola?
Which of the following represents a parabola?
Show Hint
To spot a parabola, eliminate the parameter \(t\) and reduce each pair to a relation between \(x\) and \(y\). The conic is a parabola only when the result has exactly one squared variable, i.e. it simplifies to the form \(y=ax^2+bx+c\) (or \(x=ay^2+by+c\)).
Updated On: Jun 22, 2026
- \(x=4\cos t,\ y=4\sin t\)
- \(x^2-2=-2\cos t,\ y=\cos^2\left(\frac t2\right)\)
- \(\sqrt{x}=\tan t,\ \sqrt{y}=\sec t\)
- \(x=\sqrt{1-\sin t},\ y=\sin\left(\frac t2\right)+\cos\left(\frac t2\right)\)
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The Correct Option is B
Solution and Explanation
Step 1: Use the half-angle identity.
Given in option (2), \[ y=\cos^2\left(\frac t2\right) \] We know that \[ \cos t=2\cos^2\left(\frac t2\right)-1 \] So, \[ \cos t=2y-1 \]
Step 2: Substitute in the given equation.
Also, \[ x^2-2=-2\cos t \] Substitute
Given in option (2), \[ y=\cos^2\left(\frac t2\right) \] We know that \[ \cos t=2\cos^2\left(\frac t2\right)-1 \] So, \[ \cos t=2y-1 \]
Step 2: Substitute in the given equation.
Also, \[ x^2-2=-2\cos t \] Substitute
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