Find the equation of the parabola whose focus is at $\left(1,1\right)$ and the directrix is $x - y = 3$.
Given: $\;$ Equation of directrix of parabola is $\;\;$ $x - y = 3$ $\;\;$ i.e. $\;$ $x - y -3 = 0$
Given: $\;$ Focus of parabola is at $\;$ $\left(1, 1\right)$
Let $P \left(x, y\right)$ be any point on the parabola.
Then, by definition of parabola,
Distance of $P$ from the focus $=$ distance of point $P$ from the directrix
i.e. $\;$ $\sqrt{\left(x - 1\right)^2 + \left(y - 1\right)^2} = \left|\dfrac{x - y - 3}{\sqrt{1^2 + \left(-1\right)^2}}\right|$
i.e. $\;$ $x^2 - 2x + 1 + y^2 - 2y + 1 = \dfrac{x^2 + y^2 + 9 - 2xy - 6x + 6y}{2}$
i.e. $\;$ $2x^2 - 4x + 2y^2 - 4y + 4 = x^2 + y^2 - 6x + 6y - 2xy + 9$
i.e. $\;$ $x^2 + y^2 + 2xy + 2x - 10 y - 5 = 0$ $\;\;\; \cdots \; (1)$
Equation $(1)$ is the required equation of the parabola.