Sum of the squares of the lengths of the perpendiculars of a point P on the line y = x from the lines y = 2x and y = 3x is equal to 81. `(1)/(20)OP^(2)` is equal to

Answer

To solve the problem step by step, we need to find the lengths of the perpendiculars from a point \( P(h, h) \) on the line \( y = x \) to the lines \( y = 2x \) and \( y = 3x \), and then use the given information to find \( \frac{1}{20} OP^2 \).

### Step 1: Identify the lines and point
The lines given are:
1. \( y = 2x \) (which can be rewritten as \( y - 2x = 0 \))
2. \( y = 3x \) (which can be rewritten as \( y - 3x = 0 \))

The point \( P \) on the line \( y = x \) can be represented as \( P(h, h) \).

### Step 2: Calculate the perpendicular distance from \( P(h, h) \) to the line \( y = 2x \)
The formula for the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

For the line \( y - 2x = 0 \), we have \( A = -2, B = 1, C = 0 \). Thus, the perpendicular distance \( d_1 \) from point \( P(h, h) \) is:
\[ d_1 = \frac{|-2h + h|}{\sqrt{(-2)^2 + 1^2}} = \frac{| -h |}{\sqrt{4 + 1}} = \frac{h}{\sqrt{5}} \]

### Step 3: Calculate the perpendicular distance from \( P(h, h) \) to the line \( y = 3x \)
For the line \( y - 3x = 0 \), we have \( A = -3, B = 1, C = 0 \). Thus, the perpendicular distance \( d_2 \) from point \( P(h, h) \) is:
\[ d_2 = \frac{|-3h + h|}{\sqrt{(-3)^2 + 1^2}} = \frac{| -2h |}{\sqrt{9 + 1}} = \frac{2h}{\sqrt{10}} = \frac{h\sqrt{4}}{\sqrt{10}} = \frac{h\sqrt{2}}{5} \]

### Step 4: Sum of the squares of the lengths of the perpendiculars
According to the problem, the sum of the squares of the lengths of the perpendiculars is equal to 81:
\[ d_1^2 + d_2^2 = 81 \]
Substituting \( d_1 \) and \( d_2 \):
\[ \left(\frac{h}{\sqrt{5}}\right)^2 + \left(\frac{2h}{\sqrt{10}}\right)^2 = 81 \]
This simplifies to:
\[ \frac{h^2}{5} + \frac{4h^2}{10} = 81 \]
\[ \frac{h^2}{5} + \frac{2h^2}{5} = 81 \]
\[ \frac{3h^2}{5} = 81 \]

### Step 5: Solve for \( h^2 \)
Multiplying both sides by 5:
\[ 3h^2 = 405 \]
Dividing by 3:
\[ h^2 = 135 \]

### Step 6: Calculate \( OP^2 \)
The distance \( OP \) from the origin \( O(0, 0) \) to the point \( P(h, h) \) is given by:
\[ OP^2 = h^2 + h^2 = 2h^2 \]
Substituting \( h^2 = 135 \):
\[ OP^2 = 2 \times 135 = 270 \]

### Step 7: Calculate \( \frac{1}{20} OP^2 \)
Now we find:
\[ \frac{1}{20} OP^2 = \frac{1}{20} \times 270 = \frac{270}{20} = 13.5 \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{20} OP^2 = 13.5 \]