Q107 / 180 · KCET 2024 · Mathematics
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
- Correct answerA$x y=C$
- B$x^2+y^2=C$
- C$x^2-y^2=C$
- D$\frac{y}{x}=C$
Solution
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
$\int \frac{d y}{d y}=-\int \frac{d x}{x}$
$\Rightarrow \quad \ln y=-\ln x+\ln C$
$\Rightarrow \quad \ln y=\ln \frac{C}{x}$
$\Rightarrow \quad y=\frac{C}{x}$
$\therefore \quad x y=C$
Hence, the correct answer is option: 'A'.