The parabola $ y^{2}=8 x $ and the circle $ x^{2}+y^{2}=2 $

Question

The parabola $ y^{2}=8 x $ and the circle $ x^{2}+y^{2}=2 $, Have only two common tangents which are mutually perpendicular, Have only two common tangents which are parallel to each other, Have infinitely many common tangents, Does not have any common tangent

MARKS App · Accepted Answer

Let the equation y = m x + c be the common tangents so the curve y 2 = 8 x and x 2 + y 2 = 2 Then, c = 2 m and c 2 = 2 ( 1 + m 2 ) If m 2 = t , then 4 t = 2 1 + t ⇒ t 2 + t - 2 = 0 ⇒ t + 2 t - 1 = 0 ⇒ t = 1 , - 2 Thus, m = ± 1 ( ∵ t ≠ - 2 ) Hence, tangents are y = x + c and y = - x + c which are perpendicular to each other

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