Ellipse
Equation of Ellipse
Grade 11
Question:
<p>An ellipse is drawn by taking a diameter of the circle \((x-1)^2 + y^2 = 1\) as its semi-minor axis and a diameter of the circle \(x^2 + (y-2)^2 = 4\) as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is</p>
<p>\(4x^2 + y^2 = 4\)</p>
<p>\(x^2 + 4y^2 = 8\)</p>
<p>\(4x^2 + y^2 = 8\)</p>
<p>\(x^2 + 4y^2 = 16\)</p>
Step-by-Step Solution
Key Concept: The semi-minor axis equals a radius of the first circle (not diameter), and semi-major axis equals a radius of the second circle. A diameter of a circle is 2r, so you need to extract the radii correctly and recognize which dimension becomes which for the ellipse.
<p><strong>Step 1:</strong> Identify the circles.</p><p>Circle 1: $(x-1)^2 + y^2 = 1$ has radius $r_1 = 1$, so diameter = 2</p><p>Circle 2: $x^2 + (y-2)^2 = 4$ has radius $r_2 = 2$, so diameter = 4</p><p><strong>Step 2:</strong> Interpret the semi-axes.</p><p>The semi-minor axis of the ellipse = a diameter of Circle 1 = 2, so $b = 1$</p><p>The semi-major axis of the ellipse = a diameter of Circle 2 = 4, so $a = 2$</p><p><strong>Step 3:</strong> Form the ellipse equation.</p><p>Centre at origin, axes along coordinate axes:</p><p>$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$</p><p>$$\frac{x^2}{4} + \frac{y^2}{1} = 1$$</p><p>or equivalently: $\frac{x^2}{4} + y^2 = 1$</p><p>∴ Answer: D</p>
Correct Answer: D