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 need to find the length of a focal chord for the parabola described by the equation \(y^2=8x+4y+4\) where the x-intercept of the chord is given as 3. To begin, we convert the parabola's equation into a more standard form.

Step 1: Rewrite the equation into a standard form

The given equation is:
\(y^2 = 8x + 4y + 4\)

Starting by rearranging terms, we have:
\(y^2 - 4y = 8x + 4\)

To complete the square for the \(y\)-terms, add and subtract \((2)^2 = 4\) on the left:
\((y^2 - 4y + 4) = 8x + 4 + 4\)
\((y-2)^2 = 8x + 8\)
Simplifying, we get:
\((y-2)^2 = 8(x+1)\)

This corresponds to a parabola of the form \((y-k)^2 = 4a(x-h)\), where \(a=2\), \(h=-1\), and \(k=2\).

Step 2: Analyze the focal chord

The vertex of the parabola is \((-1,2)\), and the focus at \((1,2)\) because the parameter \(4a=8\) indicates the distance \(a=2\) horizontally from the vertex.

A focal chord of a parabola is a line passing through the focus with endpoints on the parabola. Given the x-intercept of the chord is 3, we denote the chord's equation line as:

Using the point-slope formula, the slope of this line reaching from \((3,0)\) and passing through the focus \((1,2)\) is:

\( \text{slope} = \frac{2-0}{1-3} = -1 \)

The equation of the line is:
\( y-0 = -1(x-3) \) or \( y = -x + 3 \)

Step 3: Solve for the chord endpoints on the parabola

Substitute \(y=-x+3\) into the parabola's equation:
\( (-x+3-2)^2 = 8(x+1) \)
\( (-x+1)^2 = 8(x+1) \)
\( (x^2 - 2x + 1) = 8x + 8 \)
\( x^2 - 10x - 7 = 0 \)

Solving this quadratic equation:
The roots are given by \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
Here, \(a=1\), \(b=-10\), and \(c=-7\).

\( x = \frac{10 \pm \sqrt{100 + 28}}{2} = \frac{10 \pm \sqrt{128}}{2} \)
\( x = \frac{10 \pm 8\sqrt{2}}{2} \)
\( x = 5 \pm 4\sqrt{2} \)

So, the endpoints are \( (5 + 4\sqrt{2}, y_1) \) and \( (5 - 4\sqrt{2}, y_2) \).

To find \(y_1\) and \(y_2\), substitute these \(x\) values back into the linear equation \(y = -x + 3\).

Step 4: Calculate the chord length

Using the distance formula between these chord endpoints:
\(L = \sqrt{((5 + 4\sqrt{2}) - (5 - 4\sqrt{2}))^2 + ((-5 - 4\sqrt{2} + 3) - (-5 + 4\sqrt{2} + 3))^2}\)

\(L = \sqrt{(8\sqrt{2})^2 + (-8\sqrt{2})^2} = \sqrt{128 + 128} = \sqrt{256} = 16\)

The length of the focal chord is confirmed to be 16, falling perfectly within the specified range.