Question

Solve for the equation of the family of curves in which the slope is (5−𝑥)

(𝑦−3)

Determine the type of the curve and the equation of a member passing through

(2 − 1).

Answer 1

$\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{5 - x}{y - 3} \\ y - 3 \mathrm{d}y = 5 - x \mathrm{d}x\\ \int y - 3 \mathrm{d}y = \int 5 - x \mathrm{d}x\\ \frac{y^2}{2} - 3y = 5x - \frac{x^2}{2} + C\\ \textsf{At}\,\, (2, -1)\\ \frac{(-1)^2}{2} - 3(-1) = 5(2) - \frac{2^2}{2} + C\\ \frac{7}{2} = 8 + C\\ C = -\frac{9}{2}\\ \frac{y^2}{2} - 3y = 5x - \frac{x^2}{2} -\frac{9}{2}\\ x^2 + y^2 - 10x - 6y + 9 = 0\\ (x - 5)^2 + (y - 3)^2 = 25 + 9 - 9 = 25\\ \textsf{This type of curve passing}\\ \textsf{through}\,\,(2,-1)\,\, \textsf{is a circle}\\\textsf{whose centre is}\,\, (5,3)\\ \textsf{and radius is}\,\,5.$