The `1//4 :1//2 : 1//4` of TT: tt is is mathematically condesnable to the form of the bionomial expression

Options

A. (ax+by)
B. (ax-by)
C. `(ax+b)^(2)`
D. `(ax-by)^(2)`

Answer

To solve the question regarding the binomial expression that corresponds to the genotype ratio of \( \frac{1}{4} : \frac{1}{2} : \frac{1}{4} \) for the genotypes TT (homozygous tall), Tt (heterozygous tall), and tt (homozygous dwarf), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Genotypes and Their Frequencies:**
- The genotypes are:
- TT (homozygous tall)
- Tt (heterozygous tall)
- tt (homozygous dwarf)
- The given ratio \( \frac{1}{4} : \frac{1}{2} : \frac{1}{4} \) corresponds to these genotypes.

2. **Identifying the Binomial Expression:**
- In a monohybrid cross, the expected genotype ratio can be derived from the binomial expansion of \( (a + b)^2 \), where:
- \( a \) represents the dominant allele (T)
- \( b \) represents the recessive allele (t)
- The frequencies of the alleles T and t are both \( \frac{1}{2} \).

3. **Setting Up the Binomial Expression:**
- The binomial expression can be set up as:
\[ \left( \frac{1}{2}T + \frac{1}{2}t \right)^2 \]

4. **Expanding the Binomial Expression:**
- When we expand \( \left( \frac{1}{2}T + \frac{1}{2}t \right)^2 \), we apply the formula \( (x + y)^2 = x^2 + 2xy + y^2 \):
\[ \left( \frac{1}{2}T + \frac{1}{2}t \right)^2 = \left( \frac{1}{2}T \right)^2 + 2\left( \frac{1}{2}T \right)\left( \frac{1}{2}t \right) + \left( \frac{1}{2}t \right)^2 \]
- This simplifies to:
\[ \frac{1}{4}T^2 + \frac{1}{2}Tt + \frac{1}{4}t^2 \]

5. **Relating the Expanded Expression to the Genotype Ratios:**
- The coefficients of the expanded expression correspond to the genotype ratios:
- \( \frac{1}{4} \) for TT (homozygous tall)
- \( \frac{1}{2} \) for Tt (heterozygous tall)
- \( \frac{1}{4} \) for tt (homozygous dwarf)

6. **Conclusion:**
- Therefore, the binomial expression that corresponds to the genotype ratio \( \frac{1}{4} : \frac{1}{2} : \frac{1}{4} \) is:
\[ \left( \frac{1}{2}T + \frac{1}{2}t \right)^2 \]