Question

Let the two foci of an ellipse be (-1,0) and (3, 4) and the foot of perpendicular from the focus (3,4) upon a tangent to the ellipse be (4,6). Find equation of directrix

Answer 1

x + y - 10 = 0

Answer 2

A fundamental property of ellipses states that the foot of the perpendicular from a focus to any tangent line lies on the directrix corresponding to that focus. Therefore, the point $P(4, 6)$ lies on the directrix corresponding to the focus $F_2(3, 4)$.

The major axis of the ellipse passes through both foci, $F_1 = (-1, 0)$ and $F_2 = (3, 4)$. The slope of the major axis is calculated as: $m_{major} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{3 - (-1)} = \frac{4}{4} = 1$.

The directrix is always perpendicular to the major axis. Therefore, the slope of the directrix ($m_{directrix}$) is the negative reciprocal of the slope of the major axis: $m_{directrix} = -\frac{1}{m_{major}} = -\frac{1}{1} = -1$.

Since the directrix passes through the point $P(4, 6)$ and has a slope of $-1$, we can use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, to find its equation: $y - 6 = -1(x - 4)$ $y - 6 = -x + 4$ $x + y - 10 = 0$.

This is the equation of the directrix corresponding to the focus $(3, 4)$.