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 }=2 t $
$\text { hence, } \frac{2 t \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} $
$\text { from (1), } 2\left(\frac{3}{4}- a \right)=-1 \Rightarrow \frac{3}{4}- a =-\frac{1}{2} \Rightarrow a =\frac{5}{4}$