If the family of parabolas touch the pair of lines xy-2x-cy+2=0(c is 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 first need to analyze the equations provided. The lines are represented by the equation \( xy - 2x - cy + 2 = 0 \), and we need to determine the conditions under which the parabolas touch these lines while having their axes parallel to the line \( x + 2y + k = 0 \).

Understanding the Given Lines

The equation \( xy - 2x - cy + 2 = 0 \) can be rearranged to express it in a more recognizable form. By isolating \( y \), we can rewrite it as:

\( y = \frac{2x + 2}{c + x} \)

This represents a pair of lines, which can be derived from the quadratic form of the equation. The lines intersect at points that we can find by setting the determinant of the corresponding quadratic equation to zero.

Analyzing the Second Line

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

\( y = -\frac{1}{2}x - \frac{k}{2} \)

This line has a slope of \(-\frac{1}{2}\), indicating that any parabola with an axis parallel to this line will also have a slope of \(-\frac{1}{2}\). Therefore, the vertex form of the parabolas can be expressed as:

\( y = a(x - h)^2 - \frac{1}{2}(x - h) + k \)

where \( (h, k) \) is the vertex of the parabola.

Conditions for Tangency

For the parabolas to touch the lines, the discriminant of the resulting quadratic equation formed by substituting the parabola's equation into the line's equation must equal zero. This ensures that there is exactly one point of intersection, which is the condition for tangency.

Finding the Focus

The focus of a parabola defined in the vertex form \( y = a(x - h)^2 + k \) is located at the point \( (h, k + \frac{1}{4a}) \). Since the axis of the parabola is parallel to the line \( x + 2y + k = 0 \), we can infer that the focus must also lie along a line that maintains this slope.

Determining the Line for the Focus

To find the line on which the focus lies, we can substitute the coordinates of the focus into the line equation \( x + 2y + k = 0 \). This gives us:

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

which simplifies to:

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

This equation describes a relationship between \( h \), \( k \), and \( a \) that must hold true for the focus to lie on the line defined by \( x + 2y + k = 0 \).

Conclusion

In summary, the focus of the family of parabolas that touch the given lines and have axes parallel to the specified line will lie on a line defined by the relationship derived above. This line is crucial for ensuring that the parabolas maintain their tangential relationship with the lines while adhering to the specified orientation. By analyzing the conditions for tangency and the properties of parabolas, we can effectively determine the location of the focus.