Which one of the following is the common tangent to the ellipses, $\frac{{{x^2}}}{{{a^2} + {b^2}}} + \frac{{{y^2}}}{{{b^2}}}$ $=1\&$ $ \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2} + {b^2}}}$ $=1$
Which one of the following is the common tangent to the ellipses, $\frac{{{x^2}}}{{{a^2} + {b^2}}} + \frac{{{y^2}}}{{{b^2}}}$ $=1\&$ $ \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{a^2} + {b^2}}}$ $=1$
- A
$ay = bx +\sqrt {{a^4} - {a^2}{b^2} + {b^4}} $
- B
$by = ax -\sqrt {{a^4} + {a^2}{b^2} + {b^4}} $
- C
$ay = bx -\sqrt {{a^4} + {a^2}{b^2} + {b^4}} $
- D
$by = ax +\sqrt {{a^4} - {a^2}{b^2} + {b^4}} $
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Let $A = \left\{ {\left( {x,y} \right):\,y = mx + 1} \right\}$
$B = \left\{ {\left( {x,y} \right):\,\,{x^2} + 4{y^2} = 1} \right\}$
$C = \left\{ {\left( {\alpha ,\beta } \right):\,\left( {\alpha ,\beta } \right) \in A\,\,and\,\,\left( {\alpha ,\beta } \right) \in B\,\,and\,\alpha \, > 0} \right\}$ .
If set $C$ is singleton set then sum of all possible values of $m$ is
Let $A = \left\{ {\left( {x,y} \right):\,y = mx + 1} \right\}$
$B = \left\{ {\left( {x,y} \right):\,\,{x^2} + 4{y^2} = 1} \right\}$
$C = \left\{ {\left( {\alpha ,\beta } \right):\,\left( {\alpha ,\beta } \right) \in A\,\,and\,\,\left( {\alpha ,\beta } \right) \in B\,\,and\,\alpha \, > 0} \right\}$ .
If set $C$ is singleton set then sum of all possible values of $m$ is