Let $y^{2} = 4ax$ be a parabola and $x^{2} + y^{2} + 2bx = 0$ be a circle. If parabola and circle to

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Let $y^{2} = 4ax$ be a parabola and $x^{2} + y^{2} + 2bx = 0$ be a circle. If parabola and circle touch each other externally then :

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Sol. [Key Concept: External tangency of $y^{2} = 4ax$ and the circle $x^{2} + y^{2} + 2bx = 0$ fixes the signs of $a,b$ (vertex and centre on opposite sides of the contact).]The circle is $(x + b)^{2} + y^{2} = b^{2}$ (centre $( - b,0)$, through the origin).Substituting $y^{2} = 4ax$ gives $x^{2} + (4a + 2b)x = 0$.Both curves pass through the origin and touch there externally (on opposite sides) only when the parabola's axis-direction and the circle's centre are on opposite sides — i.e. $a > 0,b > 0$ or $a < 0,b < 0$ → (a), (d).

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