Let P be a point on the parabola, `y^(2)=12x` and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN,parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is `(4)/(3),` then

Options

A. `PN=4`
B. `MQ=(1)/(3)`
C. `PN=3`
D. `MQ=(1)/(4)`

Answer

To solve the problem step by step, we will analyze the given information and apply the necessary mathematical concepts.

### Step 1: Understand the Parabola
The equation of the parabola is given by \( y^2 = 12x \). This is a standard form of a parabola that opens to the right.

### Step 2: Identify Point P on the Parabola
Let \( P(t) \) be a point on the parabola. We can express the coordinates of point \( P \) in terms of a parameter \( t \):
\[ P(t) = (3t^2, 6t) \]
This is derived from the parametric equations of the parabola, where \( x = \frac{y^2}{12} \).

### Step 3: Find Point N
Point \( N \) is the foot of the perpendicular from point \( P \) to the x-axis (the axis of the parabola). Therefore, the coordinates of point \( N \) are:
\[ N = (3t^2, 0) \]

### Step 4: Find Midpoint M of PN
The midpoint \( M \) of segment \( PN \) can be calculated as:
\[ M = \left( \frac{x_P + x_N}{2}, \frac{y_P + y_N}{2} \right) = \left( \frac{3t^2 + 3t^2}{2}, \frac{6t + 0}{2} \right) = \left( 3t^2, 3t \right) \]

### Step 5: Equation of Line MN
Since the line through \( M \) is parallel to the axis of the parabola (the x-axis), it will have the same y-coordinate as \( M \):
\[ y = 3t \]

### Step 6: Find Intersection Point Q
To find the intersection point \( Q \) of the line \( y = 3t \) with the parabola \( y^2 = 12x \), we substitute \( y = 3t \) into the parabola's equation:
\[ (3t)^2 = 12x \implies 9t^2 = 12x \implies x = \frac{3t^2}{4} \]
Thus, the coordinates of point \( Q \) are:
\[ Q = \left( \frac{3t^2}{4}, 3t \right) \]

### Step 7: Find the Slope of Line NQ
The slope \( m \) of the line \( NQ \) can be calculated using the coordinates of points \( N \) and \( Q \):
\[ m = \frac{y_Q - y_N}{x_Q - x_N} = \frac{3t - 0}{\frac{3t^2}{4} - 3t^2} = \frac{3t}{\frac{3t^2}{4} - \frac{12t^2}{4}} = \frac{3t}{-\frac{9t^2}{4}} = -\frac{4}{3t} \]

### Step 8: Equation of Line NQ
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting \( N(3t^2, 0) \) and the slope \( m \):
\[ y - 0 = -\frac{4}{3t}(x - 3t^2) \implies y = -\frac{4}{3t}x + 4t \]

### Step 9: Find the y-intercept of Line NQ
The y-intercept occurs when \( x = 0 \):
\[ y = -\frac{4}{3t}(0) + 4t = 4t \]
According to the problem, the y-intercept is given as \( \frac{4}{3} \):
\[ 4t = \frac{4}{3} \implies t = \frac{1}{3} \]

### Conclusion
The value of \( t \) that satisfies the condition is \( t = \frac{1}{3} \).