Question

A circle of radius unity centred on the $y$-axis is tangent to the parabola $y=x^2$ in two places.
The coordinates of the centre of the circle are

• A. $\left(0,\frac{6}{5}\right)$
• B. $\left(0,\frac{5}{4}\right)$
• C. $\left(0,\frac{4}{3}\right)$
• D. $\left(0,\frac{3}{2}\right)$

Answer 1

Answer: (B)

${\frac{dy}{dx}}_P=2t$
$\text{ hence, }\frac{2t\left(t^2-a\right)}{t}=-1$
$\Rightarrow 2\left(t^2-a\right)=-1$ ....(1)
$\text{ also }{t}^2+{\left(t^2-a\right)}^2=1$ ......(2)
$\text{ squaring equation (1), }4{\left(t^2-a\right)}^2=1\Rightarrow 4\left(1-t^2\right)=1\Rightarrow t^2=\frac{3}{4}\Rightarrow t=\frac{\sqrt{3}}{2}$
$\text{ from (1), }2\left(\frac{3}{4}-a\right)=-1\Rightarrow \frac{3}{4}-a=-\frac{1}{2}\Rightarrow a=\frac{5}{4}$