Statement 1: An equation of a common tangent to the parabola y^2 = 16√(3)x and the ellipse 2x^2 + y^2 = 4 is y = 2x + 2√(3) .Statement 2: If the line y = mx + 4√(3)m, (m ≠ 0) is a common tangent to the parabola y^2 = 16√(3)x and the ellipse 2x^2 + y^2 = 4 , then m satisfies m^4 + 2m^2 = 24 .

For parabola y^2 = 4ax , tangent is y = mx + a/m .Here 4a = 16√(3) a = 4√(3) .Tangent: y = mx + 4√(3)m (Statement 2 part 1).For ellipse x^2/a^2 + y^2/b^2 = 1 , condition for tangency of y = mx + c is c^2 = a^2 m^2 + b^2 .The ellipse equation is x^2/2 + y^2/4 = 1 .Here a^2 = 2, b^2 = 4 .So, ≤ft( 4√(3)m )^2 = 2m^2 + 4 48/m^2 = 2m^2 + 4 24 = m^4 + 2m^2 m^4 + 2m^2 = 24 .Statement 2 is true.Solving m^4 + 2m^2 - 24 = 0 : (m^2 + 6)(m^2 - 4) = 0 m^2 = 4 m = ± 2 .If m = 2 , tangent is y = 2x + 4√(3)2 = 2

Statement 1: An equation of a common tangent to the parabola y^2 = 16√(3)x and the ellipse 2x^2 + y^2 = 4 is y = 2x + 2√(3) .Statement 2: If the line - Mathematics MCQ AIEEE 2012 2026 | ExamTest.live

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Statement 1: An equation of a common tangent to the parabola y^{2} = 163 x and the ellipse 2 x^{2} + y^{2} = 4 is y = 2 x + 23 .Statement 2: If the line y = m x + \frac{4 3}{m}, (m \neq = 0) is a common tangent to the parabola y^{2} = 163 x and the ellipse 2 x^{2} + y^{2} = 4, then m satisfies m^{4} + 2 m^{2} = 24 .

A $Statement 1 is false, statement 2 is true$
B $Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1$
C $Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1$
D $Statement 1 is true, statement 2 is false$

Solution & Step-by-step Explanation

For parabola y^{2} = 4 a x, tangent is y = m x + \frac{a}{m} .Here 4 a = 163 ⟹ a = 43 .Tangent: y = m x + \frac{4 3}{m} (Statement 2 part 1).For ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, condition for tangency of y = m x + c is c^{2} = a^{2} m^{2} + b^{2} .The ellipse equation is \frac{x ^{2}}{2} + \frac{y ^{2}}{4} = 1 .Here a^{2} = 2, b^{2} = 4 .So, (\frac{4 3}{m})^{2} = 2 m^{2} + 4 \frac{48}{m ^{2}} = 2 m^{2} + 4 ⟹ 24 = m^{4} + 2 m^{2} ⟹ m^{4} + 2 m^{2} = 24 .Statement 2 is true.Solving m^{4} + 2 m^{2} - 24 = 0 : (m^{2} + 6) (m^{2} - 4) = 0 ⟹ m^{2} = 4 ⟹ m = \pm 2 .If m = 2, tangent is y = 2 x + \frac{4 3}{2} = 2 x + 23 .So Statement 1 is true and Statement 2 explains it.

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Statement 1: An equation of a common tangent to the parabola y^{2} = 163 x and the ellipse 2 x^{2} + y^{2} = 4 is y = 2 x + 23 .Statement 2: If the line y = m x + \frac{4 3}{m}, (m \neq = 0) is a common tangent to the parabola y^{2} = 163 x and the ellipse 2 x^{2} + y^{2} = 4, then m satisfies m^{4} + 2 m^{2} = 24 .

Statement 1 is false, statement 2 is true

Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

Statement 1 is true, statement 2 is false

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Conic Sections Parabola Ellipse Common Tangent

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Statement 1: An equation of a common tangent to the parabola y2=163​x and the ellipse 2x2+y2=4 is y=2x+23​.Statement 2: If the line y=mx+m43​​,(m=0) is a common tangent to the parabola y2=163​x and the ellipse 2x2+y2=4, then m satisfies m4+2m2=24.

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