81.

The differential equation representing a family of circles touchmg the Y-axis at the origin is

• ${\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 - 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 + 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 -\hspace{0.17em}{\mathrm{y}}^2 - 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 -\hspace{0.17em}{\mathrm{y}}^2 + 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

The area of the region bounded by the curve 9x² + 4y² - 36 = 0 is

• $9\mathrm{\pi }$

• $5\mathrm{\pi }$

• $36\mathrm{\pi }$

• $6\mathrm{\pi }$

The general solution of the differential equation (2x - y + 1)dx + (2y - x + 1) dy = 0 is

• x² + y² + xy - x + y = c

• x² + y² - xy + x + y = c

• x² - y² + xy - x + y = c

• x² - y² - xy + x + y = c