The locus of the point (x, y) whose distance from the line `y=2x+2` is equal to the distance from (2, 0), is a parabola with the length of latus rectum same as that of the parabola `y=Kx^(2)`, then the value of K is equal to

Options

A. `(sqrt5)/(12)`
B. `(sqrt5)/(4)`
C. `(4)/(sqrt5)`
D. `(12)/(sqrt5)`

Answer

To solve the problem, we need to find the locus of the point (x, y) such that its distance from the line \(y = 2x + 2\) is equal to its distance from the point (2, 0).

### Step 1: Find the distance from the point (x, y) to the line \(y = 2x + 2\).

The equation of the line can be rewritten in the standard form \(Ax + By + C = 0\):
\[ 2x - y + 2 = 0 \]
Here, \(A = 2\), \(B = -1\), and \(C = 2\).

The formula for the distance \(d\) from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \(x_0 = x\) and \(y_0 = y\):
\[ d_1 = \frac{|2x - y + 2|}{\sqrt{2^2 + (-1)^2}} = \frac{|2x - y + 2|}{\sqrt{5}} \]

### Step 2: Find the distance from the point (x, y) to the point (2, 0).

The distance \(d_2\) from the point (x, y) to the point (2, 0) is given by:
\[ d_2 = \sqrt{(x - 2)^2 + (y - 0)^2} = \sqrt{(x - 2)^2 + y^2} \]

### Step 3: Set the distances equal to each other.

We set the two distances equal:
\[ \frac{|2x - y + 2|}{\sqrt{5}} = \sqrt{(x - 2)^2 + y^2} \]

### Step 4: Square both sides to eliminate the square root.

Squaring both sides gives:
\[ \frac{(2x - y + 2)^2}{5} = (x - 2)^2 + y^2 \]
Multiplying through by 5:
\[ (2x - y + 2)^2 = 5((x - 2)^2 + y^2) \]

### Step 5: Expand both sides.

Expanding the left side:
\[ (2x - y + 2)^2 = 4x^2 - 4xy + y^2 + 8x - 4y + 4 \]
Expanding the right side:
\[ 5((x - 2)^2 + y^2) = 5(x^2 - 4x + 4 + y^2) = 5x^2 - 20x + 20 + 5y^2 \]

### Step 6: Set the expanded forms equal.

Setting the two expansions equal gives:
\[ 4x^2 - 4xy + y^2 + 8x - 4y + 4 = 5x^2 - 20x + 20 + 5y^2 \]

### Step 7: Rearranging the equation.

Rearranging the equation to one side:
\[ 0 = 5x^2 - 20x + 20 + 5y^2 - 4x^2 + 4xy - y^2 - 8x + 4 \]
This simplifies to:
\[ x^2 + 5y^2 + 4xy - 28x + 24 = 0 \]

### Step 8: Identify the conic section.

This equation represents a parabola. The standard form of a parabola is \(y = kx^2\).

### Step 9: Find the length of the latus rectum.

For the parabola \(y = kx^2\), the length of the latus rectum is given by \( \frac{4}{|k|} \).

### Step 10: Relate the lengths of latus rectum.

From the derived equation, we can find the value of \(k\) by comparing the coefficients. We know that the length of the latus rectum from our derived parabola is equal to the length of the latus rectum from \(y = kx^2\).

### Step 11: Solve for \(k\).

From the equation \(4a = \frac{12}{\sqrt{5}}\), we find:
\[ k = \frac{1}{4a} = \frac{\sqrt{5}}{12} \]

Thus, the value of \(K\) is:
\[ K = \frac{1}{4a} = \frac{5}{12} \]

### Final Answer:
The value of \(K\) is \(\frac{5}{12}\).