If lx + my + n = 0 is tangent to the parabola `x^(2)=y`, them

Options

A. `t^(2)=2mn`
B. `i=4m^(2)n^(2)`
C. `m^(2)=4/n`
D. `l^(2)=4mn`

Explanation

To determine the condition under which the line \( lx + my + n = 0 \) is tangent to the parabola \( x^2 = y \), we can follow these steps:

### Step 1: Substitute the equation of the parabola into the line equation.
The equation of the parabola is given by \( y = x^2 \). We substitute this into the line equation \( lx + my + n = 0 \).

\[ lx + m(x^2) + n = 0 \]

This simplifies to:

\[ mx^2 + lx + n = 0 \]

### Step 2: Identify the quadratic form.
The equation \( mx^2 + lx + n = 0 \) is a quadratic equation in \( x \). For a line to be tangent to a parabola, this quadratic must have exactly one solution, which occurs when the discriminant is zero.

### Step 3: Calculate the discriminant.
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

In our case, \( a = m \), \( b = l \), and \( c = n \). Thus, the discriminant becomes:

\[ D = l^2 - 4mn \]

### Step 4: Set the discriminant to zero.
Since we want the line to be tangent to the parabola, we set the discriminant equal to zero:

\[ l^2 - 4mn = 0 \]

### Step 5: Rearrange the equation.
Rearranging the equation gives us the condition for tangency:

\[ l^2 = 4mn \]

### Conclusion:
Thus, the condition for the line \( lx + my + n = 0 \) to be tangent to the parabola \( x^2 = y \) is:

\[ l^2 = 4mn \]

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