The family of curves whose \(x\) and \(y\) intercepts of a tangent at any point are respectively double the \(x\) and \(y\) coordinates of that point is

Let the point on the curve be \((h, k)\).
According to the question, the equation of the tangent will be
\(\Rightarrow \quad \frac{x}{2 h}+\frac{y}{2 k}=1\)
\(\Rightarrow \quad \frac{y}{2 k}=1-\frac{y}{2 h}\)
\(\Rightarrow \quad y=-\left(\frac{2 k}{2 h}\right) x+2 k\)
So, the slope of tangent will be \(\left(-\frac{k}{h}\right)\).
Now, \(\quad \frac{d y}{d x}=-\frac{h}{k}=-\frac{y}{x}\)
\(\Rightarrow \quad \frac{d y}{y}=-\frac{d x}{x}\)
On integrating both sides, we get
\(\int \frac{d y}{d y}=-\int \frac{d x}{x}\)
\(\Rightarrow \quad \ln y=-\ln x+\ln C\)
\(\Rightarrow \quad \ln y=\ln \frac{C}{x}\)
\(\Rightarrow \quad y=\frac{C}{x}\)
\(\therefore \quad x y=C\)