A normal to the hyperbola `(x^2)/4-(y^2)/1=1` has equal intercepts on the positive x- and y-axis. If this normal touches the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` , then `a^2+b^2` is equal to (a) 5 (b) 25 (c) 16 (d) none of these

Explanation

Equation of Normal to a hyperbola is:-
`axcos(theta)+bycot(theta)=a^2+ b^2`
By given hyperbola,`a=2, b=1`

So,`2xcos(theta)+ycot(theta)=5`
As the normal makes equal intercepts on positve x & y-axis, So, `slope=-1=m`

Therefore,` (-2cos(theta)/(cot(theta)))=-1`

`Sin(theta)=1/2`

`theta=pi/6`

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