Step-by-step Solution:

To find the number of points of intersection of the ellipse \[ \frac{x^2}{4} + \frac{(y-1)^2}{9} = 1 \] and the circle \[ x^2 + y^2 = 4, \] we solve the equations simultaneously. [\]\ Rewrite the equations 1. Ellipse equation: \[ \frac{x^2}{4} + \frac{(y-1)^2}{9} = 1. \] 2. Circle equation: \[ x^2 + y^2 = 4. \] Substitute \( x^2 \) from the circle into the ellipse From the circle equation, we have: \[ x^2 = 4 - y^2. \] Substitute \( x^2 = 4 - y^2 \) into the ellipse equation: \[ \frac{4 - y^2}{4} + \frac{(y-1)^2}{9} = 1. \] Simplify: \[ \frac{4}{4} - \frac{y^2}{4} + \frac{(y-1)^2}{9} = 1. \] \[ 1 - \frac{y^2}{4} + \frac{(y-1)^2}{9} = 1. \] Eliminate \( 1 \) and simplify further \[ -\frac{y^2}{4} + \frac{(y-1)^2}{9} = 0. \] Expand \( (y-1)^2 \): \[ -\frac{y^2}{4} + \frac{y^2 - 2y + 1}{9} = 0. \] Simplify into a single quadratic equation Multiply through by \( 36 \) (the LCM of 4 and 9) to clear fractions: \[ -9y^2 + 4(y^2 - 2y + 1) = 0. \] Expand: \[ -9y^2 + 4y^2 - 8y + 4 = 0. \] Combine like terms: \[ -5y^2 - 8y + 4 = 0. \] Solve the quadratic equation \[ 5y^2 + 8y - 4 = 0. \] Use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 5 \), \( b = 8 \), \( c = -4 \). \[ y = \frac{-8 \pm \sqrt{8^2 - 4(5)(-4)}}{2(5)}. \] \[ y = \frac{-8 \pm \sqrt{64 + 80}}{10}. \] \[ y = \frac{-8 \pm \sqrt{144}}{10}. \] \[ y = \frac{-8 \pm 12}{10}. \] Calculate the roots 1. \( y = \frac{-8 + 12}{10} = \frac{4}{10} = 0.4 \). 2. \( y = \frac{-8 - 12}{10} = \frac{-20}{10} = -2. \) \[\] Find \( x \) for each \( y \) For each \( y \), use \( x^2 = 4 - y^2 \): 1. For \( y = 0.4 \): \[ x^2 = 4 - (0.4)^2 = 4 - 0.16 = 3.84. \] \( x = \pm\sqrt{3.84} \) gives two values. 2. For \( y = -2 \): \[ x^2 = 4 - (-2)^2 = 4 - 4 = 0. \] \( x = 0 \). \[\]Total points of intersection - From \( y = 0.4 \), there are 2 points.\[\] - From \( y = -2 \), there is 1 point. \[\] Total points of intersection = \( 2 + 1 = 3 \). \[\] Final Answer: Option B: 3.