If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at t…

Question

If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance of P from the origin, is :

Accepted Answer

Tangent to x2 + y2 = 4 is

y = mx $$ \pm $$ 2$$\sqrt {1 + {m^2}} $$

Also, x2 = 4y

x2 = 4mx + 8$$\sqrt {1 + {m^2}} $$

or  x2 = 4mx - 8$$\sqrt {1 + {m^2}} $$

For D = 0

we have; 16m2 + 4.8$$\sqrt {1 + {m^2}} $$ = 0

$$ \Rightarrow $$   m2 + 2$$\sqrt {1 + {m^2}} $$  = 0

$$ \Rightarrow $$    m2 = - 2$$\sqrt {1 + {m^2}} $$

$$ \Rightarrow $$   m4 = 4 + 4m2

$$ \Rightarrow $$   m4 - 4m2 - 4 = 0

$$ \Rightarrow $$    m2 = $${{4 \pm \sqrt {16 + 16} } \over 2}$$

$$ \Rightarrow $$   m2 = $${{4 \pm 4\sqrt 2 } \over 2}$$

$$ \Rightarrow $$   m2 = 2 + 2$$\sqrt 2 $$

Solutions · 1