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JEE Main 2018
Conic Sections
Parabola
Medium

Question

Given : A circle, 2x2+2y2=52{x^2} + 2{y^2} = 5 and a parabola, y2=45x{y^2} = 4\sqrt 5 x. Statement-1 : An equation of a common tangent to these curves is y=x+5y = x + \sqrt 5 . Statement-2 : If the line, y=mx+5m(m≠0)y = mx + {{\sqrt 5 } \over m}\left( {m \ne 0} \right) is their common tangent, then mm satiesfies m4−3m2+2=0{m^4} - 3{m^2} + 2 = 0.

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Solution

This problem requires us to analyze two statements concerning common tangents to a given circle and parabola. We need to determine the truthfulness of each statement and whether Statement-2 serves as a correct explanation for Statement-1.

1. Key Concepts and Formulas

  • Standard Form of a Circle: The equation of a circle centered at the origin is x2+y2=R2x^2 + y^2 = R^2, where RR is its radius.
  • Standard Form of a Parabola: The equation of a parabola symmetric about the x-axis and opening to the right is y2=4axy^2 = 4ax, where aa is a positive constant.
  • Equation of a Tangent to a Parabola: For a parabola $y^
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