1. **Problem Statement:** Find the centre, vertices, minor axis points, and foci of the ellipse given by the equation $$4x^2 + 9y^2 - 16x + 18y - 11 = 0$$ 2. **Rewrite the equation by grouping and completing the square:** $$4x^2 - 16x + 9y^2 + 18y = 11$$ Group terms: $$4(x^2 - 4x) + 9(y^2 + 2y) = 11$$ Complete the square inside each parenthesis: For $x^2 - 4x$, add and subtract $4$: $$x^2 - 4x + 4 - 4 = (x - 2)^2 - 4$$ For $y^2 + 2y$, add and subtract $1$: $$y^2 + 2y + 1 - 1 = (y + 1)^2 - 1$$ 3. **Substitute back:** $$4((x - 2)^2 - 4) + 9((y + 1)^2 - 1) = 11$$ Expand: $$4(x - 2)^2 - 16 + 9(y + 1)^2 - 9 = 11$$ Simplify constants: $$4(x - 2)^2 + 9(y + 1)^2 = 11 + 16 + 9 = 36$$ 4. **Divide both sides by 36 to get standard form:** $$\frac{4(x - 2)^2}{36} + \frac{9(y + 1)^2}{36} = 1$$ Simplify fractions: $$\frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{4} = 1$$ 5. **Identify ellipse parameters:** - Centre: $(2, -1)$ - $a^2 = 9$, so $a = 3$ (major axis radius) - $b^2 = 4$, so $b = 2$ (minor axis radius) Since $a^2 > b^2$, major axis is along the $x$-axis. 6. **Vertices:** Along $x$-axis from centre: $$ (2 \pm 3, -1) = (-1, -1) \text{ and } (5, -1) $$ 7. **Minor axis points:** Along $y$-axis from centre: $$ (2, -1 \pm 2) = (2, -3) \text{ and } (2, 1) $$ 8. **Foci:** Calculate $c$ where $c^2 = a^2 - b^2 = 9 - 4 = 5$, so $c = \sqrt{5}$. Foci along major axis ($x$-axis): $$ (2 \pm \sqrt{5}, -1) $$ --- **Summary for part (a):** - Centre: $(2, -1)$ - Vertices: $(-1, -1)$ and $(5, -1)$ - Minor axis points: $(2, -3)$ and $(2, 1)$ - Foci: $(2 - \sqrt{5}, -1)$ and $(2 + \sqrt{5}, -1)$ --- 1. **Problem Statement:** Find the centre, vertices, minor axis points, and foci of the ellipse given by the equation $$x^2 + 2y^2 - 10x + 8y + 29 = 0$$ 2. **Group and complete the square:** Group $x$ and $y$ terms: $$x^2 - 10x + 2(y^2 + 4y) = -29$$ Complete the square for $x$: $$x^2 - 10x + 25 - 25 = (x - 5)^2 - 25$$ Complete the square for $y$ inside the parentheses: $$y^2 + 4y + 4 - 4 = (y + 2)^2 - 4$$ 3. **Substitute back:** $$ (x - 5)^2 - 25 + 2((y + 2)^2 - 4) = -29 $$ Expand: $$ (x - 5)^2 - 25 + 2(y + 2)^2 - 8 = -29 $$ Simplify constants: $$ (x - 5)^2 + 2(y + 2)^2 = -29 + 25 + 8 = 4 $$ 4. **Divide both sides by 4:** $$ \frac{(x - 5)^2}{4} + \frac{2(y + 2)^2}{4} = 1 $$ Simplify: $$ \frac{(x - 5)^2}{4} + \frac{(y + 2)^2}{2} = 1 $$ 5. **Identify ellipse parameters:** - Centre: $(5, -2)$ - $a^2 = 4$, $a = 2$ - $b^2 = 2$, $b = \sqrt{2}$ Since $a^2 > b^2$, major axis is along the $x$-axis. 6. **Vertices:** Along $x$-axis from centre: $$ (5 \pm 2, -2) = (3, -2) \text{ and } (7, -2) $$ 7. **Minor axis points:** Along $y$-axis from centre: $$ (5, -2 \pm \sqrt{2}) $$ 8. **Foci:** Calculate $c$ where $c^2 = a^2 - b^2 = 4 - 2 = 2$, so $c = \sqrt{2}$. Foci along major axis ($x$-axis): $$ (5 \pm \sqrt{2}, -2) $$ --- **Summary for part (b):** - Centre: $(5, -2)$ - Vertices: $(3, -2)$ and $(7, -2)$ - Minor axis points: $(5, -2 + \sqrt{2})$ and $(5, -2 - \sqrt{2})$ - Foci: $(5 - \sqrt{2}, -2)$ and $(5 + \sqrt{2}, -2)$