Given : A circle, $$2{x^2} + 2{y^2} = 5$$ and a parabola, $${y^2} = 4\sqrt 5 x$$.
Statement-1 : An equation of a common tangent to these curves is $$y = x + \sqrt 5 $$.
Statement-2 : If the line, $$y = mx + {{\sqrt 5 } \over m}\left( {m \ne 0} \right)$$ is their common tangent, then $$m$$ satiesfies $${m^4} - 3{m^2} + 2 = 0$$.
A
Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is true; Statement-2 is false.
D
Statement-1 is false Statement-2 is true.
Explanation
Let common tangent be
$$y = mx + {{\sqrt 5 } \over m}$$
Since, perpendicular distance from center of the circle to
the common tangent is equal to radius of the circle,
An ellipse is drawn by taking a diameter of thec circle $${\left( {x - 1} \right)^2} + {y^2} = 1$$ as its semi-minor axis and a diameter of the circle $${x^2} + {\left( {y - 2} \right)^2} = 4$$ is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is:
A
$$4{x^2} + {y^2} = 4$$
B
$${x^2} + 4{y^2} = 8$$
C
$$4{x^2} + {y^2} = 8$$
D
$${x^2} + 4{y^2} = 16$$
Explanation
Equation of circle is $${\left( {x - 1} \right)^2} + {y^2} = 1$$
$$ \Rightarrow $$ radius $$=1$$ and diameter $$=2$$
$$\therefore$$ Length of semi-minor axis is $$2.$$
Equation of circle is $${x^2} + {\left( {y - 2} \right)^2} = 4 = {\left( 2 \right)^2}$$
$$ \Rightarrow $$ radius $$=2$$ and diameter $$=4$$