Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
The values of $a$ for which $y = ax^2 + ax + \frac{1}{24}$, $x = ay^2 + ay + \frac{1}{24}$ touch each other is/are
$\frac{2}{3}$
$\frac{3}{2}$
$\frac{13 + \sqrt{601}}{12}$
$\frac{13 - \sqrt{601}}{12}$
Step-by-Step Solution
Key Concept: For tangents with specified slopes meeting the curve at prescribed loci, eliminate the slope parameter using both the curve equation and tangent condition.
The point of contact lies on the line $y = x$ at $(a, a)$. The tangent slope is $\pm 1$ and relates to the curve by $a = a\sigma^2 + a\sigma + \frac{1}{24}$. Eliminating $\sigma$ from the tangent slope condition yields $\left(\frac{\pm 1 - \sigma}{2a}\right)^2 = a\left(\frac{\pm 1 - \sigma}{2a}\right) + a + \frac{1}{24}$. Solving gives $a = \frac{2}{3}, \frac{3}{2}$ and corresponding $\sigma = \frac{1}{3}, \frac{13 \pm \sqrt{601}}{12}$.
Correct Answer: 1,2,3,4