Statement 1 : An equation of a common tangent to the parabola y^{2} = 16 \sqrt{3} x and the ellipse 2x2 + y2 = 4 is y = 2x + 2 \sqrt{3}. Statement 2 : If the line y = mx + \frac{4 \sqrt{3}}{m}, (m \neq 0) is a common tangent to the parabola y^{2} = 16 \sqrt{3} x and the ellipse 2x2 + y2 = 4, then m satisfies m4 + 2m2 = 24.

Engineering

Mathematics

Tangent to Ellipse

Question

Statement 1 : An equation of a common tangent to the parabola $y^{2} = 16 \sqrt{3} x$ and the ellipse 2x² + y² = 4 is y = 2x + 2 $\sqrt{3}$ .

Statement 2 : If the line $y = mx + \frac{4 \sqrt{3}}{m}$ , (m $\neq$ 0) is a common tangent to the parabola $y^{2} = 16 \sqrt{3} x$ and the ellipse 2x² + y² = 4, then m satisfies m⁴ + 2m² = 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 false.

Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 2.

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Equation of tangent to parabola $y^{2} = 16 \sqrt{3} x$ is $y = mx + \frac{4 \sqrt{3}}{m}$ . A line y = mx + C is tangent to ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ . When c² = a²m² + b².

$\frac{16 \times 3}{m^{2}} = 2 m^{2} + 4$

m⁴ + 2m² = 24

$∴$ On solving, m² = 4 or

m² = – 6 (not possible)]

$∴$ m = ± 2

T : y = 2x + 2 $\sqrt{3}$ & y = – 2x – 2