Question

Let one end of a focal chord of the parabola \( y^2 = 16x \) be \( (16, 16) \). If \( P(\alpha, \beta) \) divides this focal chord internally in the ratio \( 5 : 2 \), then the minimum value of \( \alpha + \beta \) is equal to :

• A. 5
• B. 7
• C. 16
• D. 22

Hint:

Remember the parameter relation for focal chords: \( t_1 t_2 = -1 \). This allows you to find the coordinates of the other end immediately.

Answer 1

The Correct Option is B

To solve this problem, we need to find the focal chord of the parabola and determine the point \( P(\alpha, \beta) \) that divides this chord internally in the ratio \( 5:2 \). Let's go through the problem step-by-step. 

1. First, identify the parabola and its focus. The given equation of the parabola is \(y^2 = 16x\). This is a standard parabola of the form \(y^2 = 4ax\) where \( a = 4 \). The focus of this parabola is therefore at \((4, 0)\).
2. Calculate the equation of the focal chord. A focal chord passes through the focus of the parabola. Suppose the coordinates of the other end of the chord are \( (x_1, y_1) \). Since the chord passes through the focus \( (4, 0) \) and the given endpoint \( (16, 16) \), the slope of the chord is \(\frac{16 - 0}{16 - 4} = \frac{4}{3}\).
3. Derive the line equation for the focal chord using the point-slope form:
   • The equation is \(y - 0 = \frac{4}{3}(x - 4)\).
   • This simplifies to \(y = \frac{4}{3}x - \frac{16}{3}\).
4. To find \( P(\alpha, \beta) \), which divides the chord internally in the ratio \( 5:2 \), we use the section formula. Let the endpoints of the chord be \( A(16, 16) \) and \( B(x_1, y_1) \) with focus \( (4, 0) \).
5. Using the section formula:
   • \(\alpha = \frac{5x_1 + 2 \times 16}{7}\) and \(\beta = \frac{5y_1 + 2 \times 16}{7}\).
   • Substituting the equation of the line \(y = \frac{4}{3}x - \frac{16}{3}\) into the parabola \(y^2 = 16x\) will give the coordinates \( (x_1, y_1) \).
   • After substitution, solve the quadratic equation derived to find the other endpoint, then apply the section formula.
6. Once \( \alpha \) and \( \beta \) are calculated, find the minimum value of \( \alpha + \beta \).
7. Calculating through substitution and solving, we determine that the minimum value of \( \alpha + \beta \) is \(7\).

Thus, the correct option is \(7\).