Question

Coordinate of focus of the parabola \(4y^2 + 12x - 20y + 67 = 0\) is

• A. \((-\frac{5}{4}, \frac{17}{2})\)
• B. \((-\frac{17}{2}, \frac{5}{4})\)
• C. \((-\frac{17}{4}, \frac{5}{2})\)
• D. \((-\frac{5}{2}, \frac{17}{4})\)

Hint:

When converting the general equation of a conic to its standard form, always remember to balance the equation carefully after completing the square. A common mistake is forgetting to multiply the added term by the factor outside the parenthesis.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question requires us to find the coordinates of the focus of a given parabola. To do this, we first need to convert the given equation into the standard form of a parabola.

Step 2: Key Formula or Approach:
The standard equation of a parabola that opens horizontally is \((y - k)^2 = 4a(x - h)\) (opens right) or \((y - k)^2 = -4a(x - h)\) (opens left).
The vertex of such a parabola is at \((h, k)\).
The focus for a parabola opening leftwards is at \((h - a, k)\).

Step 3: Detailed Explanation:
The given equation of the parabola is: \[ 4y^2 + 12x - 20y + 67 = 0 \] To convert it to the standard form, we need to complete the square for the terms involving \(y\).
First, group the \(y\) terms and move the other terms to the right side: \[ 4y^2 - 20y = -12x - 67 \] Factor out the coefficient of \(y^2\) from the left side: \[ 4(y^2 - 5y) = -12x - 67 \] Now, complete the square inside the parenthesis. We take half of the coefficient of \(y\) (\(-5\)), square it, and add it inside the parenthesis. Half of \(-5\) is \(-\frac{5}{2}\), and its square is \(\frac{25}{4}\). \[ 4(y^2 - 5y + \frac{25}{4}) = -12x - 67 + 4(\frac{25}{4}) \] Note that we added \(4 \times \frac{25}{4} = 25\) to the right side to balance the equation.
Simplify both sides: \[ 4(y - \frac{5}{2})^2 = -12x - 67 + 25 \] \[ 4(y - \frac{5}{2})^2 = -12x - 42 \] Divide the entire equation by 4 to isolate the squared term: \[ (y - \frac{5}{2})^2 = \frac{-12x - 42}{4} \] \[ (y - \frac{5}{2})^2 = -3x - \frac{21}{2} \] Factor out the coefficient of \(x\) on the right side: \[ (y - \frac{5}{2})^2 = -3(x + \frac{7}{2}) \] Now, we compare this with the standard form \((y - k)^2 = -4a(x - h)\).
By comparison, we have: \[ k = \frac{5}{2} \] \[ h = -\frac{7}{2} \] \[ -4a = -3 \implies a = \frac{3}{4} \] The vertex of the parabola is \((h, k) = (-\frac{7}{2}, \frac{5}{2})\).
Since the equation is of the form \((y-k)^2 = -4a(x-h)\), the parabola opens to the left.
The coordinates of the focus are given by \((h - a, k)\).
Substituting the values of \(h\), \(a\), and \(k\): \[ \text{Focus} = (-\frac{7}{2} - \frac{3}{4}, \frac{5}{2}) \] To subtract the x-coordinates, we find a common denominator: \[ -\frac{7}{2} - \frac{3}{4} = -\frac{14}{4} - \frac{3}{4} = -\frac{17}{4} \] So, the coordinates of the focus are \((-\frac{17}{4}, \frac{5}{2})\).

Step 4: Final Answer:
The calculated coordinates of the focus are \((-\frac{17}{4}, \frac{5}{2})\), which matches option (C).