From a variable point P, a pair of tangents are drawn to the ellipse $x^2+4y^2=4,$ the point of contact being Q and R. Let the sum of ordinates of the points Q and R be unity. If the locus of P has the equation $x^2+4y^2=4Ry,$ then the value of R is _____________ 

Let the variable point P is $(\alpha ,\beta )$  
$Q(x_1,y_1) R(x_2,y_2)$ are two points on the ellipse $\frac{x^2}{4}+y^2=1$  
Mid point of QR is $\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}$ 
Given $y_1+y_2=1$ 
Equation of chord QR is $s_1=s_{11}$ is $(x_1,y_1)$ it mid point of chord
$\Rightarrow \frac{x\left(\frac{x_1+x_2}{2}\right)}{4}+\frac{y\left(\frac{1}{2}\right)}{1}-1=\frac{{\left(\frac{x_1+x_2}{2}\right)}^2}{4}+\frac{{\left(\frac{1}{4}\right)}^2}{1}-1$ 
$\Rightarrow \frac{x(x_1+x_2)}{2}+4y\left(\frac{1}{2}\right)={\left(\frac{x_1+x_2}{2}\right)}^2+1$  ________(1)
From $p(\alpha ,\beta )$ chord of contact QR equation is $\frac{\alpha x}{4}+\frac{\beta y}{1}-1=0$ ________ (2)
1 and 2 represents same line
$\frac{\frac{x_1+x_2}{2}}{\left(\frac{\alpha }{4}\right)}=\frac{2}{\beta }=\frac{-{\left(\frac{x_1+x_2}{2}\right)}^2+1}{-1}$ 
 $\frac{x_1+x_2}{2}=\frac{\alpha }{2\beta },{\left(\frac{x_1+x_2}{2}\right)}^2+1=\frac{2\beta }{ }$
${\left(\frac{\alpha }{2\beta }\right)}^2+1=\frac{2}{\beta }\Rightarrow {\alpha }^2+4{\beta }^2=8\beta$ 
locus of $(\alpha ,\beta )$ is $x^2+4y^2=8y$  _________(3)
comparing $x^2+y^2=4Ry$ with (3)
R = 2