Two parabolas with a common vertex and with axes along x-axis and y-axis respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is:

Options

A. `4(x+y)+3=0`
B. `8(2x+y)+3=0`
C. `3(x+y)+4=0`
D. `x+2y+3=0`

Answer

To find the equation of the common tangent to the two parabolas with a common vertex and axes along the x-axis and y-axis, we will follow these steps:

### Step 1: Identify the Parabolas
Given that the length of the latus rectum of each parabola is 3, we can determine the parameters of the parabolas.

- For the parabola with the axis along the x-axis, the equation is given by:
\[ y^2 = 4ax \]
The length of the latus rectum is given by \(4a = 3\), which implies:
\[ a = \frac{3}{4} \]

- For the parabola with the axis along the y-axis, the equation is:
\[ x^2 = 4by \]
Similarly, the length of the latus rectum is given by \(4b = 3\), which implies:
\[ b = \frac{3}{4} \]

### Step 2: Write the Tangent Equation
The general form of the tangent to the parabola \(y^2 = 4ax\) is given by:
\[ y = mx + \frac{a}{m} \]
Substituting \(a = \frac{3}{4}\):
\[ y = mx + \frac{3/4}{m} = mx + \frac{3}{4m} \]

For the parabola \(x^2 = 4by\), the tangent equation is:
\[ x = my + \frac{b}{m} \]
Substituting \(b = \frac{3}{4}\):
\[ x = my + \frac{3/4}{m} = my + \frac{3}{4m} \]

### Step 3: Set the Tangents Equal
Since we are looking for a common tangent, we set the two equations equal to each other:
\[ mx + \frac{3}{4m} = my + \frac{3}{4m} \]

### Step 4: Rearranging the Equation
Rearranging gives:
\[ mx - my = 0 \implies m(x - y) = 0 \]
Since \(m \neq 0\) (as we are looking for a non-horizontal tangent), we have:
\[ x - y = 0 \implies x = y \]

### Step 5: Find the Equation of the Tangent
Now, we can substitute \(x = y\) into either tangent equation. Using the tangent equation for \(y^2 = 4ax\):
\[ y = mx + \frac{3}{4m} \]
Substituting \(y = x\):
\[ x = mx + \frac{3}{4m} \]
Rearranging gives:
\[ x - mx = \frac{3}{4m} \implies x(1 - m) = \frac{3}{4m} \]

### Step 6: Solve for the Tangent Equation
To find the equation of the tangent, we can express it in the standard form:
\[ 4mx - 4y + 3 = 0 \]
This is the equation of the common tangent.

### Final Step: Simplifying
To express the equation in a more standard form, we can multiply through by \(4\) to eliminate the fraction:
\[ 4x + 4y + 3 = 0 \]

Thus, the final equation of the common tangent to the two parabolas is:
\[ 4x + 4y + 3 = 0 \]