Question

If the line \( 2x - 3y + c = 0 \) passes through the focus of the parabola \( x^2 = -8y \), then the value of \( c \) is equal to:

• A. 4
• B. -6
• C. 6
• D. -4
• E. 2

Hint:

To find the value of \( c \) when a line passes through the focus of a parabola, substitute the coordinates of the focus into the equation of the line.

Answer 1

The Correct Option is B

Step 1:Understand the geometry of the problem.
The equation of the given parabola is \( x^2 = -8y \). This is a standard form of a parabola that opens downwards. The general form of a parabola \( x^2 = 4ay \) has its focus at \( (0, -a) \), where \( a \) is the distance from the vertex to the focus. In our case, the equation is \( x^2 = -8y \), so \( 4a = 8 \) and \( a = 2 \). Thus, the focus of the parabola is at \( (0, -2) \).
Step 2:Equation of the line.
We are given the equation of the line as \( 2x - 3y + c = 0 \), which can be rewritten as: \[ y = \frac{2}{3}x + \frac{c}{3} \] The line passes through the focus of the parabola, which is at \( (0, -2) \).
Step 3:Substitute the focus coordinates into the equation of the line.
Substitute \( x = 0 \) and \( y = -2 \) into the equation of the line: \[ -2 = \frac{2}{3}(0) + \frac{c}{3} \] Simplifying: \[ -2 = \frac{c}{3} \] Multiplying both sides by 3: \[ c = -6 \]
Step 4:Final Answer.
Therefore, the value of \( c \) is \( -6 \).