Question:
The focus of the parabola \( y^2 - 4y - x + 3 = 0 \) is
The focus of the parabola \( y^2 - 4y - x + 3 = 0 \) is
Show Hint
Always complete the square first to convert parabola into standard form.
Updated On: May 1, 2026
- \( \left(\frac{3}{4}, 2\right) \)
- \( \left(\frac{3}{4}, -2\right) \)
- \( (2, \frac{3}{4}) \)
- \( \left(-\frac{3}{4}, 2\right) \)
- \( (2, -\frac{3}{4}) \)
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The Correct Option is A
Solution and Explanation
Concept:
Standard form:
\[
(y-k)^2 = 4a(x-h)
\]
Focus:
\[
(h+a, k)
\]
Step 1: Rearrange equation.
\[ y^2 - 4y = x - 3 \]
Step 2: Complete square.
\[ (y-2)^2 - 4 = x - 3 \Rightarrow (y-2)^2 = x +1 \]
Step 3: Compare with standard form.
\[ (y-2)^2 = 4a(x+1) \Rightarrow 4a = 1 \Rightarrow a = \frac{1}{4} \]
Step 4: Find focus.
\[ (h,k) = (-1,2) \] \[ \text{Focus} = (-1 + \tfrac{1}{4}, 2) = \left(-\frac{3}{4}, 2\right) \]
Step 1: Rearrange equation.
\[ y^2 - 4y = x - 3 \]
Step 2: Complete square.
\[ (y-2)^2 - 4 = x - 3 \Rightarrow (y-2)^2 = x +1 \]
Step 3: Compare with standard form.
\[ (y-2)^2 = 4a(x+1) \Rightarrow 4a = 1 \Rightarrow a = \frac{1}{4} \]
Step 4: Find focus.
\[ (h,k) = (-1,2) \] \[ \text{Focus} = (-1 + \tfrac{1}{4}, 2) = \left(-\frac{3}{4}, 2\right) \]
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