Two straight lines pass through the fixed points `(+-a, 0)` and have slopes whose products is `pgt0` Show that the locus of the points of intersection of the lines is a hyperbola.

Explanation

Let the equation of the lines be
`y=m_(1)(x-a)`
`and y=m_(2)(x+a)`
`therefore" "m_(1)m_(2)=p`
`therefore" "y^(2)=m_(1)m_(2)(x^(2)-a^(2))=p(x^(2)-a^(2))`
Hence, the locus of the points of intersection is
`y^(2)=p(x^(2)-a^(2))`
`" "px^(2)-y^(2)=pa^(2)`
which is a hyperbola.