Question:
<p>The position of the point (1, 3) with respect to the ellipse 4x<sup>2</sup> + 9y<sup>2</sup> - 16x - 54y + 61 = 0 is</p>
<p style="display:inline">on the major axis</p>
<p style="display:inline">on the ellipse</p>
<p style="display:inline">on the minor axis</p>
<p style="display:inline">outside the ellipse</p>
Step-by-Step Solution
Key Concept: The position of a point relative to an ellipse is determined by the sign of the expression $S_1$ obtained by substituting the point's coordinates into the ellipse equation $S=0$, where $S_1 < 0$ signifies the point is inside.
<p>Here, 4 + 9(3)<sup>2</sup> - 16(1) - 54(3) + 61 < 0<br />
Therefore, the point is inside the ellipse<br />
<span class="math-tex">$\frac{4(x-2)^{2}}{36}+\frac{9(y-3)^{2}}{36}=1$</span><br />
Now, equation of major axis is y - 3 = 0 and point (1, 3) lies on it.</p>
Correct Answer: A