Question

If the line 2x-3y+c=0 passes through the focus of the parabola x² = -8y, then the value of c is equal to

• A. 4
• B. -6
• C. 6
• D. -4
• E. 2

Answer 1

The Correct Option is B

The equation of the given parabola is \( x^2 = -8y \).
This is a standard form of the parabola \( x^2 = 4ay \), where the focus is at \( (0, -a) \).
Comparing \( x^2 = -8y \) with \( x^2 = 4ay \), we find \( 4a = -8 \), so \( a = -2 \).
Therefore, the focus of the parabola is at \( (0, -2) \).
The line is given by the equation \( 2x - 3y + c = 0 \).
For this line to pass through the focus \( (0, -2) \), substitute \( x = 0 \) and \( y = -2 \) into the line's equation: \[ 2(0) - 3(-2) + c = 0 \] \[ 0 + 6 + c = 0 \] \[ c = -6 \]

The correct option is (B) : \(-6\)

Answer 2

The equation of the parabola is \(x^2 = -8y\). We can rewrite this as \(x^2 = 4(-2)y\).

Comparing this with the standard equation of a parabola \(x^2 = 4ay\), we have \(4a = -8\), so \(a = -2\).

For a parabola of the form \(x^2 = 4ay\), the focus is at the point \((0, a)\). In our case, \(a = -2\), so the focus of the parabola \(x^2 = -8y\) is at \((0, -2)\).

The line 2x - 3y + c = 0 passes through the focus (0, -2). Substituting these coordinates into the equation of the line, we get:

\(2(0) - 3(-2) + c = 0\)

\(0 + 6 + c = 0\)

Therefore, \(c = -6\).