Question Bank: w283
Determine the differential equation of a family of lines passing through $(h, k)$.
- $(y-k)dx - (x-h)dy = 0$
- $(y-k)dx + (x-h)dy = 0$
- $(x-h)dx - (y-k)dy = 0$
- $(x-h)dx + (y-k)dy = 0$
All lines through the fixed point $(h, k)$ differ only in slope $m$, so eliminate $m$ as the arbitrary constant.
Point-slope form:
$y - y_1 = m(x - x_1)$
$y - k = m(x - h)$
Replace $m$ with $\dfrac{dy}{dx}$:
$y - k = \dfrac{dy}{dx}(x - h)$
$(y - k)\,dx = (x - h)\,dy$
$\boxed{(y-k)dx - (x-h)dy = 0}$
Point-slope form:
$y - y_1 = m(x - x_1)$
$y - k = m(x - h)$
Replace $m$ with $\dfrac{dy}{dx}$:
$y - k = \dfrac{dy}{dx}(x - h)$
$(y - k)\,dx = (x - h)\,dy$
$\boxed{(y-k)dx - (x-h)dy = 0}$