Question

The equation of the line passing through the points \( \left( ct_1, \frac{c}{t_1} \right) \) and \( \left( ct_2, \frac{c}{t_2} \right) \) is

• A. \( x + t_1 t_2 y = c(t_1 + t_2) \)
• B. \( y + t_1 t_2 x = c(t_1 + t_2) \)
• C. \( x - t_1 t_2 y = c(t_1 + t_2) \)
• D. \( y - t_1 t_2 x = c(t_1 + t_2) \)

Hint:

To find the equation of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \), first calculate the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Then use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \). Substitute the given coordinates and simplify the equation to match one of the provided options.

Answer 1

The Correct Option is A

Let the two points be \( P_1 = \left( ct_1, \frac{c}{t_1} \right) \) and \( P_2 = \left( ct_2, \frac{c}{t_2} \right) \).
The slope of the line passing through \( P_1 \) and \( P_2 \) is: $$ m = \frac{\frac{c}{t_2} - \frac{c}{t_1}}{ct_2 - ct_1} = \frac{c \left( \frac{t_1 - t_2}{t_1 t_2} \right)}{c(t_2 - t_1)} = \frac{c(t_1 - t_2)}{c t_1 t_2 (t_2 - t_1)} = \frac{-(t_2 - t_1)}{t_1 t_2 (t_2 - t_1)} = -\frac{1}{t_1 t_2} $$ The equation of the line passing through \( (x_1, y_1) \) with slope \( m \) is \( y - y_1 = m(x - x_1) \).
Using point \( P_1 \left( ct_1, \frac{c}{t_1} \right) \) and slope \( m = -\frac{1}{t_1 t_2} \): $$ y - \frac{c}{t_1} = -\frac{1}{t_1 t_2} (x - ct_1) $$ Multiply by \( t_1 t_2 \): $$ t_1 t_2 y - ct_2 = -x + ct_1 $$ Rearranging the terms: $$ x + t_1 t_2 y = ct_1 + ct_2 $$ $$ x + t_1 t_2 y = c(t_1 + t_2) $$ This matches option (A).