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

To find the value of \( c \) in the line equation \( 2x - 3y + c = 0 \) such that it passes through the focus of the parabola \( x^2 = -8y \), we need to follow these steps:

1. Identify the focus of the parabola: 
   The equation of the given parabola is \( x^2 = -8y \). 
   This equation is in the standard form \( x^2 = -4ay \), where \( -4a = -8 \). From this, we solve for \( a \): \(a = \frac{8}{4} = 2\). Hence, the focus of the parabola is \( (0, -2) \).
2. Substitute the focus into the line equation:
   We substitute \( x = 0 \) and \( y = -2 \) into the line equation \( 2x - 3y + c = 0 \): \(2(0) - 3(-2) + c = 0\)
   This simplifies to: \(6 + c = 0\)
3. Solve for \( c \):
   By solving the equation \( 6 + c = 0 \), we get: \(c = -6\)
4. Verify the options:
   Among the given options, the correct value is \( -6 \).

Therefore, the value of \( c \) is \(-6\).