Intersection of tangent of parabola with directix and tangent at vertex

Question

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!

Answer 1

Key property: The portion of tangent between point of contact and the point where it meets directrix subtends a right angle at the focus.

Proof: Let ($ at^2, 2at$) be a point on $y^2=4ax$ with focus S. Let Q be the point where tangent at P meets directrix x= -a.

Equation of tangent is $ty=x+at^2$ Q is ($-a,\frac{-a}{t}+at$)

Slope of SP is $\frac{2at}{at^2-a}$ Slope of SQ is $\frac{\frac{-a}{t}+at}{-a-a}$

Clearly, $m_1×m_2= -1,$ hence perpendicualr.

Also, image of focus with respect to a tangent lies on directrix. So, we proved that the quadrilateral in figure is cyclic (opposite angles supplementary).

Answer 2

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:

• The dashed segment and $AS$ have the same length
• The tangent line bisects the angle between the dashed segment and $AS$

By SAS, the triangles on the two sides of $AB$ are congruent, and thus angle $ASB$ is right.