<p>The equation \((26x - 1)^2 + (26y - 3)^2 = k(5x - 12y + 1)^2\) will represent a parabola if <i>k</i> is equal to:</p>
Step-by-Step Solution
Key Concept: A conic equation represents a parabola when the distance from a point to a fixed point (focus) equals the distance from that point to a fixed line (directrix). The equation given is in the form: distance² to focus = k × (distance to directrix)², which represents a parabola when k = 1.
<p><strong>Step 1:</strong> Recognize the form of the equation. The equation $(26x - 1)^2 + (26y - 3)^2 = k(5x - 12y + 1)^2$ is in the form where the left side represents the distance squared from point $(x,y)$ to the point $(\frac{1}{26}, \frac{3}{26})$, and the right side involves the distance squared from the point to a line.</p><p><strong>Step 2:</strong> Identify the focus and directrix. The left side $(26x - 1)^2 + (26y - 3)^2$ can be written as $676(x - \frac{1}{26})^2 + 676(y - \frac{3}{26})^2 = 26^2[(x - \frac{1}{26})^2 + (y - \frac{3}{26})^2]$. The focus is at $F(\frac{1}{26}, \frac{3}{26})$.</p><p><strong>Step 3:</strong> Express the distance to the directrix line. The distance from point $(x,y)$ to the line $5x - 12y + 1 = 0$ is $d = \frac{|5x - 12y + 1|}{\sqrt{5^2 + (-12)^2}} = \frac{|5x - 12y + 1|}{\sqrt{25 + 144}} = \frac{|5x - 12y + 1|}{13}$. Therefore, $d^2 = \frac{(5x - 12y + 1)^2}{169}$.</p><p><strong>Step 4:</strong> Rewrite the equation. $(26x - 1)^2 + (26y - 3)^2 = k(5x - 12y + 1)^2$ can be expressed as $26^2[(x - \frac{1}{26})^2 + (y - \frac{3}{26})^2] = k(5x - 12y + 1)^2$, which gives $676 \cdot d_{F}^2 = k(5x - 12y + 1)^2$.</p><p><strong>Step 5:</strong> Apply the parabola condition. For a parabola, the distance to focus equals the distance to directrix: $d_F = d_{directrix}$. This means $d_F^2 = d_{directrix}^2$. Comparing with our equation: $676 \cdot d_F^2 = k \cdot 169 \cdot d_F^2$, we get $k = \frac{676}{169} = 4$.</p><p><strong>∴ Answer:</strong> r</p>
Correct Answer: r