Explanation
Solution
1. Find the equation of the parabola f(x)
By the definition of a parabola, any point (x,y) on the parabola is equidistant from the focus and the directrix.
(x+21)2+(y−0)2=∣y−(−21)∣
Squaring both sides:
So, f(x)=x2+x.
2. Analyze the domain of the set S
The equation given is:
tan−1(f(x))+sin−1(f(x)+1)=2π
For the square roots and the inverse sine function to be defined:
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f(x)≥0 (because of f(x))
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f(x)+1≥0 (which is already covered if f(x)≥0)
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f(x)+1≤1 (domain of sin−1(z) is [−1,1])
From f(x)+1≤1, we get:
Since we also need f(x)≥0, the only possible value for f(x) is:
3. Check if f(x)=0 satisfies the original equation
If f(x)=0:
tan−1(0)+sin−1(0+1)=tan−1(0)+sin−1(1)
The equation holds true.
4. Find the values of x
We solve f(x)=0:
Thus, S={0,−1}. The set S contains exactly two elements.
Correct Option: (1) contains exactly two elements
