Question

The centre of the circle passing through the point $(0,1)$ and touching the curve $y=x^2$ at $\left(2,4\right)$ is

• A. $\left[\frac{-16}{5},\frac{27}{10}\right]$
• B. $\left[\frac{-16}{7},\frac{53}{10}\right]$
• C. $\left[\frac{-16}{5},\frac{53}{10}\right]$
• D. None of these

Answer 1

The correct option is C

$\left[\frac{-16}{5},\frac{53}{10}\right]$

Explanation for the correct option:

Finding the center of the circle:

The general equation of the circle is $x^2+y^2+2gx+2fy+c=0\dots \left(1\right)$.

Given, the circle passes through point $\left(0,1\right)$.

Substitute $x=0$ and $y=1$ in equation $\left(1\right)$.

$$\begin{gathered} \Rightarrow 1+2f+c=0 \\ \Rightarrow c=-1-2f \end{gathered}$$

Substitute the value of $c$ in equation $\left(1\right)$.

So, equation of circle become $x^2+y^2+2gx+2fy-2f-1=0\dots \left(2\right)$

Circle also passes through $\left(2,4\right)$, that means this point satisfy the circle.

Substitute $x=2$ and $y=4$.in equation $\left(2\right)$.

$$\begin{gathered} \Rightarrow 4+16+4g+8f-2f-1=0 \\ \Rightarrow 4g+6f+19=0\dots \left(3\right) \end{gathered}$$

The center of the circle should satisfy equation $\left(3\right)$.

Checking for Option (C).

$\Rightarrow 4\left(\frac{-16}{5}\right)+6\left(\frac{53}{10}\right)+19=0$

Option (C) satisfies the equation.

Explanation for the incorrect option :

For Option (A),

$\Rightarrow 4\left(\frac{-16}{5}\right)+6\left(\frac{27}{10}\right)+19\ne 0$

since the equation is not satisfied, it is incorrect.

For Option (B),

$\Rightarrow 4\left(\frac{-16}{7}\right)+6\left(\frac{53}{10}\right)+19\ne 0$

since the equation is not satisfied, it is incorrect.

Therefore, the correct answer is option (C).