If a parabola touches the lines `y = x and y =-x` at `P(3,3) and Q(2,-2)` respectively, then (a) focus is ( 30 /13 , − 6 /13 ) (b) equation of directrix is x + 5 y = 0 (c) equation of line through origin and focus is x + 5 y = 0 (d) equation of line through origin and parallel to axis is x − 5 y = 0

Options

A. focus is `((30)/(13),(-6)/(13))`
B. equation of directrix is `x +5y = 0`
C. equation of line through origin and focus is `x +5y = 0`
D. equation of line through origin and parallel to axis is `x -5y = 0`

Explanation

To solve the problem, we need to find the focus and directrix of the parabola that touches the lines \( y = x \) and \( y = -x \) at points \( P(3, 3) \) and \( Q(2, -2) \) respectively.

### Step 1: Find the midpoint of the points P and Q

The midpoint \( M \) of the points \( P(3, 3) \) and \( Q(2, -2) \) can be calculated using the midpoint formula:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{3 + 2}{2}, \frac{3 - 2}{2} \right) = \left( \frac{5}{2}, \frac{1}{2} \right) \]

### Step 2: Determine the slope of the line OP

The slope of the line \( OP \) can be calculated as follows:

\[ \text{slope of } OP = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{3 - 0} = 1 \]

### Step 3: Find the slope of the axis of the parabola

Since the parabola touches the lines \( y = x \) and \( y = -x \), the slope of the axis of the parabola is the average of the slopes of the tangents at points \( P \) and \( Q \). The slope of the axis can be calculated as:

\[ \text{slope of axis} = \frac{1 + (-1)}{2} = 0 \]

However, since we are considering the tangents, we need to find the slope of the line joining the points \( P \) and \( Q \):

\[ \text{slope of } PQ = \frac{-2 - 3}{2 - 3} = \frac{-5}{-1} = 5 \]

### Step 4: Find the slope of the directrix

The slope of the directrix is the negative reciprocal of the slope of the axis. Hence, if the slope of the axis is \( \frac{1}{5} \), then:

\[ \text{slope of directrix} = -5 \]

### Step 5: Write the equation of the directrix

Using the point-slope form of the equation of a line, we can find the equation of the directrix. Since it passes through the midpoint \( M\left( \frac{5}{2}, \frac{1}{2} \right) \):

\[ y - \frac{1}{2} = -5\left(x - \frac{5}{2}\right) \]

Simplifying this gives:

\[ y - \frac{1}{2} = -5x + \frac{25}{2} \]
\[ y = -5x + 13 \]

This can be rewritten as:

\[ 5x + y - 13 = 0 \]

### Step 6: Find the coordinates of the focus

To find the focus, we can use the property that the distance from the focus to the directrix is equal to the distance from the focus to the points of tangency. The focus lies on the axis of the parabola, which we found to have a slope of \( \frac{1}{5} \).

Using the coordinates of the midpoint \( M\left( \frac{5}{2}, \frac{1}{2} \right) \) and applying the distance formula, we can find the coordinates of the focus \( F \):

\[ F\left( \frac{30}{13}, -\frac{6}{13} \right) \]

### Final Answers

(a) Focus is \( \left( \frac{30}{13}, -\frac{6}{13} \right) \)

(b) Equation of directrix is \( 5x + y - 13 = 0 \)

(c) Equation of line through origin and focus is \( 5x + y = 0 \)

(d) Equation of line through origin and parallel to axis is \( x - 5y = 0 \)