Given: A circle, 2x^2 + 2y^2 = 5 and a parabola, y^2 = 4√(5)x .Statement - I: An equation of a common tangent to these curves is y = x + √(5) .Stateme - Mathematics MCQ JEE Main 2026 | ExamTest.live

mediumMCQJEE Main2026Mathematics

1 mark

Given: A circle, 2 x^{2} + 2 y^{2} = 5 and a parabola, y^{2} = 45 x .Statement - I: An equation of a common tangent to these curves is y = x + 5 .Statement - II: If the line y = m x + \frac{5}{m} (m \neq = 0) is their common tangent, then m satisfies m^{4} - 3 m^{2} + 2 = 0 .

A $Statement - I is True; Statement - II is True; Statement - II is not a correct explanation for Statement - I$
B $Statement - I is True; Statement - II is False.$
C $Statement - I is False; Statement - II is True$
D $Statement - I is True; Statement - II is True; Statement - II is a correct explanation for Statement - I$

Solution & Step-by-step Explanation

1. Let y = m x + \frac{a}{m} be a tangent to y^{2} = 4 a x . Here a = 5, so y = m x + \frac{5}{m} .2. For this to be tangent to the circle x^{2} + y^{2} = \frac{5}{2}, the distance from origin (0, 0) must equal radius \frac{5}{2} . \frac{∣ \frac{5}{m} ∣}{1 + m ^{2}} = \frac{5}{2} \frac{5}{m ^{2} ( 1 + m ^{2} )} = \frac{5}{2} \Rightarrow m^{2} (1 + m^{2}) = 2 \Rightarrow m^{4} + m^{2} - 2 = 0 (m^{2} + 2) (m^{2} - 1) = 0 \Rightarrow m^{2} = 1 \Rightarrow m = \pm 1 3. For m = 1, tangent is y = x + 5 . (Statement I is True).4. Statement II says m^{4} - 3 m^{2} + 2 = 0, but the correct condition is m^{4} + m^{2} - 2 = 0 . Thus Statement II is False.

Try it yourself before checking the explanation above.

Given: A circle, 2 x^{2} + 2 y^{2} = 5 and a parabola, y^{2} = 45 x .Statement - I: An equation of a common tangent to these curves is y = x + 5 .Statement - II: If the line y = m x + \frac{5}{m} (m \neq = 0) is their common tangent, then m satisfies m^{4} - 3 m^{2} + 2 = 0 .

Statement - I is True; Statement - II is True; Statement - II is not a correct explanation for Statement - I

Statement - I is True; Statement - II is False.

Statement - I is False; Statement - II is True

Statement - I is True; Statement - II is True; Statement - II is a correct explanation for Statement - I

Practice with timed mock tests and track your performance across Mathematics.