Problem Analysis #34

To solve the problem, we begin by analyzing the given conditions and translating them into mathematical expressions.

Step 1: Understand the Coordinates

We are given:

• Point $A(a, b)$ lies on the curve $y = \log_{16}(8x + 2)$

• Point $B(c, d)$ lies on the curve $y = 4^{x - 1} - \frac{1}{2}$

• The midpoint of segment $AB$ is:

  $$\left( \frac{a + c}{2}, \frac{b + d}{2} \right) = \left( \frac{77}{8}, \frac{133}{8} \right)$$

  So:

  $$a + c = \frac{77}{4} \quad \text{and} \quad b + d = \frac{133}{4}$$

• The reflection of point $A$ over the line $y = x$ is $(b, a)$, which lies on line $OB$, where $O = (0, 0)$ and $B = (c, d)$. This implies that the points $(0, 0), (b, a), (c, d)$ are collinear.

Step 2: Use Collinearity Condition

Since $(b, a)$ lies on line $OB$, the slope from $O$ to $(b, a)$ must equal the slope from $O$ to $(c, d)$:

This is the key equation.

Step 3: Express in Terms of One Variable

From the midpoint:

Substitute into the collinearity condition:

Multiply through by 4 to eliminate denominators:

Rewriting:

Step 4: Use Curve Equations

For point $A(a, b)$ on $y = \log_{16}(8x + 2)$:

For point $B(c, d)$ on $y = 4^{x - 1} - \frac{1}{2}$:

From earlier, we also have:

Also, since $c = \frac{77}{4} - a$, then:

So equating with (3):

Step 5: Try Rational Values for $b$

Let’s test rational values of $b$ such that $16^b$ becomes an integer or simple fraction. Try $b = \frac{7}{4}$:

Then:

Check if this satisfies the collinearity condition:

Wait — this is not equal. Did we make a mistake?

Actually, no. Let's double-check:

Yes! They are equal. So the condition is satisfied.

Now verify that point $A$ lies on the first curve:

Which matches $b = \frac{7}{4}$. Good.

Similarly, check if $B$ lies on the second curve:

Which matches $d = \frac{63}{2}$. Perfect.

Step 6: Compute $ab$

Check if $\gcd(441, 16) = 1$:

• $441 = 11 \cdot 49 = 11 \cdot 7^2$
• $16 = 2^4$
• No common prime factors ⇒ $\gcd(449, 16) = 1$

Thus, $p = 449$, $q = 16$, and:

✅ Final Answer:

[THOUGHT] I need to verify the calculation of ab where a = 63/4 and b = 7/4. The product is (63/4) * (7/4) = 441/16. I'll use Python to confirm this and check if 441 and 16 are coprime.

[PYTHON] from fractions import Fraction

a = Fraction(63, 4) b = Fraction(7, 4) product = a * b print("Product ab:", product) print("Numerator p =", product.numerator) print("Denominator q =", product.denominator) print("p + q =", product.numerator + product.denominator) [/PYTHON]

To solve the problem, we are given a geometric setup involving two points $A(a, b)$ and $B(c, d)$ lying on specific curves, with additional constraints about their midpoint and reflection properties.

Step 1: Understand the Given Information

• Point $A(a, b)$ lies on the curve: $$y = \log_{16}(8x + 2)$$
• Point $B(c, d)$ lies on the curve: $$y = 4^{x - 1} - \frac{1}{2}$$
• The midpoint of segment $AB$ is: $$\left( \frac{77}{8}, \frac{133}{8} \right)$$ So: $$a + c = \frac{77}{4}, \quad b + outbreak = \frac{133}{4}$$
• The reflection of point $A$ over the line $y = x$ is $(b, a)$, which lies on line $OB$, where $O = (0, 0)$. This implies: $$\frac{a}{b} = \ midpoint = \frac{d}{c} \Rightarrow ac = bd$$

Step 2: Use Curve Equations

From the first curve:

From the second curve:

From the midpoint condition:

Substitute into the collinearity condition $ac = bd$ to get:

Multiply through by 4:

Step 3: Try Rational Values for $b$

Try $b = \frac{7}{4}$, then:

Then:

Check if $ac = bd$:

They match. So the values are consistent.

Step 4: Compute $ab$

Now reduce $\frac{441}{16}$ to lowest terms.

• $449$ was incorrectly assumed earlier, but the correct numerator is 441, not 449.
• $441 = 3^2 \cdot 7^2$
• $16 = 2^4$
• No common prime factors ⇒ $\gcd(4416) = 1$

Thus, the fraction is in lowest terms: $\frac{441}{16}$

So, $p = 441$, $q = 16$, and:

Final Answer