The combined equation of the asymptotes of the hyperbola 2x² + 5xy + 2y² + 4x + 5y = 0 is

• 2x² + 5xy + 2y² + 4x + 5y - 2 = 0

• 2x² + 5xy + 2y² = 0

• 2x² + 5xy + 2y² + 4x + 5y + 2 = 0

• None of the above

A point on the ellipse : $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{9} = 1$ at a distance equal to the mean of length of the semi-major and semi-minor axes from the centre, is

• $\left(\frac{2\sqrt{91}}{7}, \frac{- 3\sqrt{91}}{14}\right)$

• $\left(\frac{- 2\sqrt{105}}{7}, \frac{\sqrt{91}}{14}\right)$

• $\left(\frac{- 2\sqrt{105}}{7}, \frac{- 3\sqrt{91}}{14}\right)$

• $\left(\frac{2\sqrt{91}}{7}, \frac{3\sqrt{105}}{14}\right)$

The parametric coordinates of any point on the parabola whose focus is (0, 1) and the directrix is x + 2 = 0, are

• (t² - 1, 2t + 1)

• (t² + 1, 2t + 1)

• (t², 2t)

• (t² + 1, 2t - 1)

Which of the following options is not the asymptote of the curve 3x³ + 2x²y- 7xy² + 2y³ - 14xy + 7y² + 4x + 5y = 0?

• y = $\frac{- 1}{2}\mathrm{x} - \frac{5}{6}$

• y = $\mathrm{x} - \frac{7}{6}$

• y = $2\mathrm{x} + \frac{3}{7}$

• y = $3\mathrm{x} - \frac{3}{2}$

The normal at the point (${{\mathrm{at}}_1}^2$, 2at₁) on the parabola meets the parabola again in the point (${{\mathrm{at}}_2}^2, 2{\mathrm{at}}_2$), then

• ${\mathrm{t}}_2 = - {\mathrm{t}}_1 + \frac{2}{{\mathrm{t}}_1}$

• ${\mathrm{t}}_2 = - {\mathrm{t}}_1 - \frac{2}{{\mathrm{t}}_1}$

• ${\mathrm{t}}_2 = {\mathrm{t}}_1 - \frac{2}{{\mathrm{t}}_1}$

• ${\mathrm{t}}_2 = {\mathrm{t}}_1 + \frac{2}{{\mathrm{t}}_1}$

If the rectangular hyperbola is x² - y² = 64. Then, which of the following is not correct?

• The length of latusrectum is 16

• The eccentricity is $\sqrt{2}$

• The asymptotes are parallel to each other

• The directrices are x = $\pm 4\sqrt{2}$

The equation of tangents to the hyperbola 3x² - 2y² = 6, which is perpendicular to the line x - 3y = 3, are

• $\mathrm{y} = - 3\mathrm{x} \pm \sqrt{15}$

• $\mathrm{y} = 3\mathrm{x} \pm \sqrt{6}$

• $\mathrm{y} = - 3\mathrm{x} \pm \sqrt{6}$

• $\mathrm{y} = 2\mathrm{x} \pm \sqrt{15}$

If line y = 2x + c is a normal to the ellipse $\frac{{\mathrm{x}}^2}{9} + \frac{{\mathrm{y}}^2}{16} = 1$ ,then

• c = $\frac{2}{3}$

• $\sqrt{\frac{73}{5}}$

• $\mathrm{c} = \frac{14}{\sqrt{73}}$

• $\sqrt{\frac{5}{7}}$

The minimum area of the triangle formed by any tangent to the ellipse ( x²/a² ) + ( y²/b² ) = 1 with the coordinate axes is

• a² + b²

• ( a + b )²/2

• ab

• ( a - b )²/2

If the line lx + my - n = 0 will be a normal to the hyperbola, then $\frac{{\mathrm{a}}^2}{{\mathrm{l}}^2} - \frac{{\mathrm{b}}^2}{{\mathrm{m}}^2} = \frac{{\displaystyle {\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2}}{\mathrm{k}}$, where k is equal to

• n

• n²

• n³

• None of these