Animated Solution for Mathematics - Circles: The centre of the circle passing through the point $(0,1)$ and touching the curve $y=x^2$ at $(2,4)$ is

To find the center of the circle, we first identify the geometry at the point of contact $P(2,4)$ on the parabola $y=x^2$. The slope of the tangent line at any point is given by the derivative:

At $P(2,4)$, where $x=2$, the slope of the tangent $m_t$ is:

The center of the circle must lie on the normal line, which is perpendicular to the tangent at $P$. The slope of the normal $m_n$ is the negative reciprocal of $m_t$:

Using the point-slope form $y-y_1=m(x-x_1)$ for the normal line passing through $P(2,4)$, we have:

Multiplying by $4$ and rearranging terms, we obtain the equation of the normal line:

Let the center of the circle be $C(h,k)$. Since $C$ lies on the normal line, its coordinates must satisfy:

The circle passes through $A(0,1)$ and $P(2,4)$. Since both points lie on the circle, the distance from the center $C$ to $A$ must equal the distance from $C$ to $P$ ($C{A}^2=C{P}^2$):

Expanding both sides, we get:

Canceling $h^2$ and $k^2$ from both sides and simplifying the linear terms leads to:

We now solve the system of equations: 1) $h=18-4k$ 2) $4h+6k=19$

Substituting the first into the second:

Finally, solving for $h$:

The center of the circle is $(-\frac{16}{5},\frac{53}{10})$.