Question

If the x-intercept of a focal chord of the parabola \(y^2=8 x+4 y+4\) is 3 , then the length of this chord is equal to ___

Hint:

For parabolas, the length of a focal chord can be directly calculated using 16a, where a is the parameter defining the parabola in the form y² = 4ax.

Answer 1

Correct Answer: 16

We are given the equation \(y^2 = 8x + 4y + 4\).
We can rewrite this equation by completing the square for the \(y\) terms:
\[y^2 - 4y = 8x + 4\]
\[y^2 - 4y + 4 = 8x + 4 + 4\]
\[(y - 2)^2 = 8(x + 1).\]
This equation represents a parabola. We can compare it to the standard form of a parabola, \(Y^2 = 4aX\), where:
   \(a = 2\),
   \(X = x + 1\),
   \(Y = y - 2\).
The vertex of the parabola in the \(XY\) plane is \((0, 0)\). In the \(xy\) plane, the vertex is obtained by setting \(X = 0\) and \(Y = 0\), which gives \(x = -1\) and \(y = 2\).
The focus of the parabola in the \(XY\) plane is \((a, 0)\), which is \((2, 0)\). In the \(xy\) plane, the focus is found by setting \(X = 2\) and \(Y = 0\), resulting in \(x + 1 = 2 \implies x = 1\) and \(y - 2 = 0 \implies y = 2\). Therefore, the focus is \((1, 2)\).
Focal Chord
We are given a point \((3, 0)\) on the parabola. A line passing through the focus \((1, 2)\) can be written as:
\[y - 2 = m(x - 1),\]
where \(m\) is the slope of the line.
Since the point \((3, 0)\) lies on this line, we can substitute its coordinates into the equation:
\[0 - 2 = m(3 - 1)\]
\[-2 = 2m \implies m = -1.\]
So, the equation of the focal chord is:
\[y - 2 = -1(x - 1), \quad \text{or } y = -x + 3.\]
Conclusion
The length of the focal chord is \(16\).