Question

Let \( y^2 = 16x \), from point \( (16, 16) \) a focal chord is passing. Point \( (\alpha, \beta) \) divides the focal chord in the ratio 2:3, then the minimum value of \( \alpha + \beta \) is:

Hint:

For problems involving focal chords of parabolas, use the properties of the parabola's parametric equations and the section formula to find the dividing point and solve for the required quantities.

Answer 1

Correct Answer: 11

Step 1: Understand the given parabola equation.
The equation of the parabola is given as \( y^2 = 16x \), which is a standard equation of a parabola with vertex at \( (0, 0) \) and focus at \( (4, 0) \). The point \( (16, 16) \) lies on the parabola and also lies on the focal chord of the parabola. A focal chord is a line passing through the focus that intersects the parabola at two points. The property of a focal chord is that the product of the \( x \)-coordinates of the points of intersection is constant and equal to the length of the semi-latus rectum of the parabola, which is 4 for the given parabola.
Step 2: Use the property of the focal chord.
The equation of a focal chord can be given by the relation between the points on the parabola. If \( P(\alpha, \beta) \) divides the focal chord in the ratio \( 2:3 \), we can use the parametric form of the parabola to express the coordinates of the points on the parabola. Let the parametric equations of the parabola be: \[ x = t^2, \quad y = 4t \] where \( t \) is the parameter corresponding to the point \( (x, y) \) on the parabola.
Step 3: Find the parameter corresponding to \( (16, 16) \).
Substitute \( x = 16 \) and \( y = 16 \) into the parametric equations: \[ 16 = t^2 \quad \text{and} \quad 16 = 4t \] Solving for \( t \), we find \( t = 4 \). Thus, the point \( (16, 16) \) corresponds to \( t = 4 \).
Step 4: Find the coordinates of the point dividing the focal chord.
The coordinates of the point dividing the focal chord in the ratio \( 2:3 \) can be calculated using the section formula. Since the ratio is \( 2:3 \), the parametric form of the division gives the value of \( t \) for the dividing point: \[ t = \frac{3 \times 4 + 2 \times 0}{2 + 3} = \frac{12}{5} \] Substitute this value of \( t \) into the parametric equations for \( x \) and \( y \): \[ x = t^2 = \left( \frac{12}{5} \right)^2 = \frac{144}{25} \] \[ y = 4t = 4 \times \frac{12}{5} = \frac{48}{5} \] Thus, the coordinates of the point dividing the focal chord are \( \left( \frac{144}{25}, \frac{48}{5} \right) \).
Step 5: Find the sum \( \alpha + \beta \).
The minimum value of \( \alpha + \beta \) is: \[ \alpha + \beta = \frac{144}{25} + \frac{48}{5} \] \[ \alpha + \beta = \frac{144}{25} + \frac{240}{25} = \frac{384}{25} = 11 \] Thus, the minimum value of \( \alpha + \beta \) is 11.