1. Find the foci, vertices and sketch each of the following ellipses: \( \begin{array}{ll}\text { (a) } \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 & \text { (b) } 4 x^{2}+y^{2}=1 .\end{array} \) 2. Find the equation of the ellipse centred at the origin with vertical focal axis if semimajor aris is 5 and the distance between the foci and vertices is 1 unit.

**1. (a) Ellipse:** \( \frac{x^2}{25}+\frac{y^2}{9}=1 \) - **Center:** The center is at the origin \( (0,0) \). - **Semiaxes:** Since the denominator under \( x^2 \) is bigger, \[ a^2 = 25 \quad \text{and} \quad b^2 = 9, \] so the semimajor axis is \( a=5 \) and the semiminor axis is \( b=3 \). - **Vertices:** The vertices lie along the \( x \)-axis, at \[ (\pm a,\,0) = (\pm5,\,0). \] - **Foci:** The focal distance \( c \) is given by \[ c^2 = a^2 - b^2 = 25 - 9 = 16 \quad \Longrightarrow \quad c=4. \] Thus, the foci are at \[ (\pm c,\,0) = (\pm4,\,0). \] --- **1. (b) Ellipse:** \( 4x^2+y^2=1 \) - **Rewrite in standard form:** Divide by 1 to write \[ \frac{x^2}{\frac{1}{4}}+\frac{y^2}{1}=1. \] In this form, since \( 1 > \frac{1}{4} \), the major axis is vertical. - **Semiaxes:** Here, \[ a^2 = 1 \quad \text{and} \quad b^2 = \frac{1}{4}, \] so \( a=1 \) and \( b=\frac{1}{2} \). - **Vertices:** With a vertical major axis, the vertices are at \[ (0, \pm a) = (0, \pm1). \] - **Foci:** The focal distance \( c \) is found using \[ c^2 = a^2 - b^2 = 1 - \frac{1}{4}=\frac{3}{4} \quad \Longrightarrow \quad c=\frac{\sqrt{3}}{2}. \] So the foci are at \[ (0, \pm c) = \left(0, \pm \frac{\sqrt{3}}{2}\right). \] --- **2. Ellipse with vertical focal axis, semimajor axis \( a=5 \), and distance between a vertex and the corresponding focus equal to 1** Since the ellipse is centered at the origin with a vertical major axis: - **Vertices:** They are at \[ (0, \pm a) = (0, \pm5). \] - **Distance from vertex to focus:** The distance between a vertex \( (0, \pm5) \) and the corresponding focus \( (0, \pm c) \) is \[ a-c = 1. \] Thus, \[ c=5-1=4. \] - **Find \( b \):** For an ellipse with a vertical major axis, \[ c^2 = a^2 - b^2. \] Substituting, \[ 4^2 = 5^2 - b^2 \quad \Longrightarrow \quad 16 = 25 - b^2 \quad \Longrightarrow \quad b^2 = 25 - 16 = 9. \] So, \( b=3 \). - **Equation of the ellipse:** The standard form for a vertical ellipse is \[ \frac{x^2}{b^2}+\frac{y^2}{a^2}=1, \] hence, \[ \frac{x^2}{9}+\frac{y^2}{25}=1. \]