Tangents are drawn to ---$3x^2-2y^2=6$ from a point P. If these tangents intersect the coordinate axes at concyclic points then what is the locus of P?
Here is my approach:
I took the general slope format of the tangent as: $y=mx ± \sqrt{(2m^2-3)}$
and taking P as: (h,k), adjusting and squaring the equation:
$(h^2-2)m^2 - 2hkm + k^2 +3$ -(i) .....[as (h,k) lies on the tangent]
This is where I got stuck. When I viewed my book,
It took the slope of these tangents as $m_1$ and $m_2$ and then took: $m_1.m_2=1$
After this it was simple to take in values from the equation (i) but what I did not understand is why they took $m_1.m_2=1$
Can someone please explain this to me?
Tangents intersect $y$ axis at $y=q_1$, $y=q_2$, and intersect $x$ axis at $x=-q_1/m_1$, $x=-q_2/m_2$. Intersecting chords theorem implies then $$ q_1q_2={q_1q_2\over m_1m_2}, \quad\text{that is:}\quad m_1m_2=1. $$