I saw this question from Advanced Porblems in Coordinate Geometry by Vikas Gupta for JEE Advanced pertaining to Conic Section.
If the line $x + y −1 = 0$ is a tangent to a parabola with focus $(1, 2)$
at $A$ and intersects the directrix at $B$ and tangent at vertex at $C$
respectively, then $AC, BC$ is equal to :
We can solve it by forming an equation of parabola, finding slope etc. (takes about an A4 page) but that is not my concern here.
When I checked upon the authors solution to see if my approach was correct, I was amazed!
This is how the he solved the problem.
And then, the most beautiful I thing I have ever seen…$$BC*AC=(CS)^2$$
So my doubt is how was he so certain that the circle would pass exactly through the points $A, B$ and $S$. Is it some sort of a property or an axiom or some weird result?
I thought for a while, but nothing seems to strike. Any help would be appreciated. Many thanks!
it can be seen that the line through the focus and the projection of the given point onto the directrix is perpendicular to the tangent to the given point. Here, these two important points are $(0,6)$ and $(8,0)$ respectively and the line through them has slope $\frac{-6}8=-\frac34$. So the tangent we seek has slope $\frac43$.
Best Answer
The key is to show that angle $ASB$ is right, meaning the center of the circle through those points lies on $AB$. We do this using the dashed line segment and two known properties of the parabola:
By SAS, the triangles on the two sides of $AB$ are congruent, and thus angle $ASB$ is right.