If the family of parabolas touch the pair of lines xy-2x-cy=2(where c is a constant) with axis parallel to the line x+2y+k=0, then its focus lies on the line?

To solve the problem regarding the family of parabolas that touch the given pair of lines, we need to analyze the equations and their relationships carefully. The lines are defined by the equation \(xy - 2x - cy = 2\), and we want to find out where the focus of the parabolas lies when their axes are parallel to the line defined by \(x + 2y + k = 0\).

Understanding the Given Lines

The first step is to rewrite the equation of the lines in a more manageable form. The equation \(xy - 2x - cy = 2\) can be rearranged to express \(y\) in terms of \(x\):

• Rearranging gives us \(y(x - c) = 2x + 2\).
• Thus, \(y = \frac{2x + 2}{x - c}\), which indicates that we have a hyperbola rather than two distinct lines unless \(c\) takes specific values.

Identifying the Orientation of the Parabolas

The second line, \(x + 2y + k = 0\), can be rearranged to find its slope:

• Rearranging gives \(y = -\frac{1}{2}x - \frac{k}{2}\).
• This line has a slope of \(-\frac{1}{2}\), indicating that the axis of the parabolas will also have this slope, meaning they open either upwards or downwards.

Finding the Focus of the Parabolas

For a parabola with a vertical axis, the standard form is given by:

y = a(x - h)^2 + k

Here, \((h, k)\) represents the vertex of the parabola. The focus of a parabola is located at \((h, k + \frac{1}{4a})\). Since the axis of the parabolas is parallel to the line \(x + 2y + k = 0\), we can determine that the focus will also have a specific relationship with the line.

Determining the Line on Which the Focus Lies

Given that the focus lies on a line parallel to \(x + 2y + k = 0\), we can express the line on which the focus lies. The general form of a line parallel to this would be:

x + 2y + m = 0

for some constant \(m\). Therefore, the focus of the parabolas, which is of the form \((h, k + \frac{1}{4a})\), must satisfy this equation:

h + 2(k + \frac{1}{4a}) + m = 0

Conclusion

In summary, the focus of the family of parabolas that touch the given pair of lines will lie on a line of the form \(x + 2y + m = 0\), where \(m\) is a constant determined by the specific parameters of the parabolas. This relationship stems from the orientation of the parabolas and the characteristics of the lines they interact with.