Animated Solution for Mathematics - Conic Sections: Let $x^2+y^2+Ax+By+C=0$ be a circle passing through $(0,6)$ and touching the parabola $y=x^2$ at $(2,4)$. Then $A+C$ is equal to ____.

Imagine you are standing on a coordinate plane. You see a parabola, $y=x^2$, curving gracefully upward.

Now, imagine a circle, $x^2+y^2+Ax+By+C=0$, that doesn't just cross this parabola—it kisses it. It touches the parabola perfectly at the point $P(2,4)$.

This moment of contact is not just a collision; it is a geometric alignment. When two curves touch, they share a common tangent at that point. This is the secret key to unlocking this problem.

The Parabola's Secret

First, let's focus on the parabola $y=x^2$. We need the tangent line at the point $(2,4)$.

Using the $T=0$ method, we replace $y$ with $\frac{y+y_1}{2}$ and $x^2$ with $x\cdot x_1$. At $(2,4)$, this becomes:

Simplifying this, we get $y+4=4x$, or $4x-y-4=0$. This is the equation of the common tangent line. It is the bridge between our two curves.

The Circle's Language

Now, let's turn our attention to the circle. We are given the general equation $x^2+y^2+Ax+By+C=0$.

We apply the same $T=0$ method to find the tangent at $(2,4)$. The equation becomes:

To make this manageable, we multiply the entire equation by $2$, yielding:

Expanding this, we get $4x+8y+Ax+2A+By+4B+2C=0$. Grouping the terms by $x$ and $y$, we arrive at:

The Elegant Cancellation

Here is where the magic happens. We have two equations for the same line: $4x-y-4=0$ and $(A+4)x+(B+8)y+(2A+4B+2C)=0$.

Because they represent the same line, their coefficients must be proportional. We write:

We need to find $A+C$. If we look at the second and third ratios, we see something beautiful:

Cross-multiplying gives us $4(B+8)=2A+4B+2C$. Expanding the left side, we get $4B+32=2A+4B+2C$.

Notice how the $4B$ terms cancel out perfectly from both sides! We are left with $32=2A+2C$. Dividing by $2$, we get the final result:

We found the answer without ever needing the point $(0,6)$. In the high-stakes environment of the JEE, you will often encounter distractors designed to make you doubt your path.

Trust the geometry, trust the algebra, and stay focused on the goal. The beauty of this problem lies in how the variables align to reveal the truth, leaving the unnecessary information behind.