Animated Solution for Mathematics - Conic Sections: Comprehension Passage

The ellipse is defined by the equation:

Given the external point $P(3,4)$, we draw two tangents to the ellipse touching at points $A$ and $B$. The line segment $AB$ is the Chord of Contact.

Using the elegant $T=0$ theorem, we substitute $P(3,4)$ into the ellipse equation to find the equation of the chord:

This linear equation serves as the backbone for our geometric analysis.

Finding Beauty in Verticality

We consider $△PAB$ with vertices $P(3,4)$, $A(3,0)$, and $B(-\frac{9}{5},\frac{8}{5})$.

Since $P(3,4)$ and $A(3,0)$ share the same $x$-coordinate, the side $PA$ is a vertical line. Consequently, the altitude dropped from vertex $B$ to $PA$ must be a horizontal line.

Because this altitude passes through $B(-\frac{9}{5},\frac{8}{5})$, its equation is simply:

Next, we find the altitude from $P$ to the line $AB$. Since the slope of $AB$ ($x+3y=3$) is $-\frac{1}{3}$, the slope of the perpendicular altitude is $3$.

Using the point-slope form for $P(3,4)$:

Solving the system $y=\frac{8}{5}$ and $y=3x-5$, we find the orthocenter $H$:

Thus, the orthocenter is $H(\frac{11}{5},\frac{8}{5})$.

The Parabola's Secret

We seek the locus of a point equidistant from the focus $P(3,4)$ and the directrix $AB$ ($x+3y-3=0$). By definition, this locus is a parabola.

We set the squared distance to the focus equal to the squared distance to the directrix:

Expanding both sides, we obtain:

After careful algebraic simplification, we arrive at the final equation of the locus: