Animated Solution for Mathematics - Conic Sections: Comprehension Passage

A Journey Through Parameters

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a problem; we are exploring the elegant, rigid, and beautiful world of conic sections.

The parabola $y^2=4ax$ is one of the most fundamental shapes in physics and mathematics, and today, we will dissect its properties with the precision of a surgeon.

The Focal Chord Mystery

Imagine you are standing on the coordinate plane, looking at the parabola $y^2=4ax$. We have a point $P$ defined by the parameter $t$, sitting at $(a{t}^2,2at)$.

We also have a focal chord $PQ$ passing through the focus $F(a,0)$. For any focal chord, the parameters of the two endpoints are linked by the relation:

Because the line connecting $P$ and $Q$ must pass through the focus, we equate the slopes of $PF$ and $QF$ to derive this relationship. Thus, the coordinates of $Q$ are fixed as:

The Parallelism Constraint

Next, we are introduced to a point $K(2a,0)$ and a point $R(a{r}^2,2ar)$. We are told that the line $PK$ is parallel to the line $QR$.

Using the coordinates $P(a{t}^2,2at)$ and $K(2a,0)$, the slope $m_{PK}$ is:

For the line $QR$, using $Q\left(\frac{a}{t^2},-\frac{2a}{t}\right)$ and $R(a{r}^2,2ar)$, the slope $m_{QR}$ is:

Equating these two slopes, $\frac{2}{r-1/t}=\frac{2t}{t^2-2}$, we find the value of $r$:

The Tangent-Normal Intersection

Now, we shift our focus to a point $S(a{s}^2,2as)$ on the parabola, given the condition $st=1$, which implies $s=1/t$. We seek the intersection of the tangent at $P$ and the normal at $S$.

The equation of the tangent at $P(t)$ is $ty=x+a{t}^2$, which we rewrite as:

The equation of the normal at $S(s)$ is $y+sx=2as+a{s}^3$. Substituting $x=ty-a{t}^2$ and $s=1/t$ into this equation, we obtain:

Expanding and simplifying the expression, we get:

To isolate $y$, we move $at$ to the right side and combine terms over the common denominator $t^3$:

Recognizing the numerator as a perfect square, $a(t^2+1{)}^2$, we arrive at the final ordinate:

The algebra has collapsed into a beautiful, compact result. You have successfully navigated the geometry, the slopes, and the intersection of these curves, revealing the underlying simplicity of the problem.