`ypropx^(2)`.It is known that y=10 for a particular value of x. Find the value of y when the value of x is halved.

Options

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Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Understand the relationship
We know that \( y \) is directly proportional to \( x^2 \). This can be expressed mathematically as:
\[ y = k \cdot x^2 \]
where \( k \) is a constant of proportionality.

### Step 2: Find the constant \( k \)
We are given that \( y = 10 \) for a particular value of \( x \). Let's denote that particular value of \( x \) as \( a \). Substituting these values into the equation gives:
\[ 10 = k \cdot a^2 \]
From this, we can solve for \( k \):
\[ k = \frac{10}{a^2} \]
Let's label this as Equation (1).

### Step 3: Find \( y \) when \( x \) is halved
Now, we need to find the value of \( y \) when \( x \) is halved, i.e., when \( x = \frac{a}{2} \). We substitute \( x = \frac{a}{2} \) into the original equation:
\[ y = k \cdot \left(\frac{a}{2}\right)^2 \]
This simplifies to:
\[ y = k \cdot \frac{a^2}{4} \]

### Step 4: Substitute \( k \) from Equation (1)
Now we substitute the value of \( k \) from Equation (1):
\[ y = \left(\frac{10}{a^2}\right) \cdot \frac{a^2}{4} \]
The \( a^2 \) terms cancel out:
\[ y = \frac{10}{4} = \frac{5}{2} \]

### Conclusion
Thus, the value of \( y \) when \( x \) is halved is:
\[ \boxed{\frac{5}{2}} \]