A line $y = 2x+5$ intersects a hyperbola only at a point $(-2,1)$. The equation of one of its asymptotes is $3x+2y+1=0$. If hyperbola passes through the point $(-1,0)$ then what is the equation of hyperbola?
This is a question I found, and the solution has given the following equation as the equation of the hyperbola: $(2x-y+\lambda)(3x+2y+1)+k = 0$. Where did they get this equation from? I just know the basics of what an asymptote is, it is a line that touches the hyperbola at infinity, or basically a tangent at infinity. Where did they get this equation from?