Show that the angle that a tangent line of a hyperbola forms with the intersection with a vertical line, and a focus is a right angle.

Question

Question: Let $P(p, q)$ be a first quadrant point on $x^2/a^2 −y^2/b^2 = 1$. Let $D$ be the point where the tangent at $P$ meets the line $x =a^2c$. $F_2$ is the focus $(c, 0)$.

Show that $∠DF_2P$ is a right angle.

I believe the way to solve this problem is to confirm that the slope of $DP$ is the negative reciprocal of the slope of $DF_2$.But the points being $(p,q)$, $(a^2,y)$. and $(c,0)$ doesn't seem to get me there.

You mean the line $x=a^2/c$ (which is the directrix), I suppose. This is a famous property of the conics. — Intelligenti pauca, Mar 03 '21 at 08:52

score 0 · Answer 1 · answered Mar 03 '21 at 05:45

Hint: Try using parametric coordinates $( a\sec \theta ,b\tan \theta )$ for the point P.

Henceforth calculate the intersection of the tangent line $\frac{x\sec \theta }{a} -\frac{y\tan \theta }{b} =1$ with the line $x=a^{3} e$, assuming the focus of the hyperbola is (ae,0), where e is the eccentricity.

This intersection point turns out to be: $\left( a^{3} e,\frac{\left( a^{2}\sec \theta -1\right) b}{\tan \theta }\right)$.

So, we have P, D and F. Can you now confirm that the slope of DP is the negative reciprocal of the slope of DF?

Show that the angle that a tangent line of a hyperbola forms with the intersection with a vertical line, and a focus is a right angle.

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