Question:
<p>If foci of an ellipse are (0, ±3) and length of semimajor axis is 5 units, then find the equation of ellipse.</p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{x}{8}\right)^2+\left(\frac{y}{10}\right)^2=1\)</span></p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{x}{4}\right)^2+\left(\frac{y}{5}\right)^2=1\)</span></p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{x}{10}\right)^2+\left(\frac{y}{8}\right)^2=1\)</span></p>
<p style="display:inline"><span class="math-tex">\(\left(\frac{x}{5}\right)^2+\left(\frac{y}{4}\right)^2=1\)</span></p>
Step-by-Step Solution
Key Concept: Identify the ellipse's vertical orientation from the foci on the y-axis and use the focal distance relationship $b^2 = a^2 - c^2$ to determine the semi-minor axis.
<p><span class="math-tex">\(\left(\frac{x}{4}\right)^2+\left(\frac{y}{5}\right)^2=1\)</span><br />
Given, <span class="math-tex">\(a=5\)</span> and <span class="math-tex">\(c=3=>b^2=a^2-c^2=5^2-3^2=4^2=>b=4\)</span>.<br />
Equation of ellipse with major axis along <span class="math-tex">\(y\)</span>-axis is <span class="math-tex">\(\left(\frac{x}{b}\right)^2+\left(\frac{y}{a}\right)^2=1\)</span>.<br />
So, equation of given ellipse is <span class="math-tex">\(\left(\frac{x}{4}\right)^2+\left(\frac{y}{5}\right)^2=1\)</span>.</p>
Correct Answer: B