If the normal at one end of the latusrectum of an ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ passes through the one end of the minor axis, then

• e⁴ - e² + 1 = 0

• e² - e + 1 = 0

• e² + e + 1 = 0

• e⁴ + e² - 1 = 0

152.

The equation of the curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact, is

• a parabola

• an ellipse

• a circle

• a hyperbola

The solution of cos(x + y)dy = dx, is

• y = $\tan \left(\frac{\mathrm{x} + \mathrm{y}}{2}\right) + \mathrm{C}$

• $\mathrm{y} = {\cos }^{-1}\left(\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right)$

• $\mathrm{y} = \sec \left(\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right) + \mathrm{C}$

• None of the above

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)

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 = $\frac{2}{3}$

• $\sqrt{\frac{73}{5}}$

• $\mathrm{c} = \frac{14}{\sqrt{73}}$

• $\sqrt{\frac{5}{7}}$