Question

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:

• A. $xy = C$
• B. $x^2 + y^2 = C$
• C. $x^2 - y^2 = C$
• D. $\frac{y}{x} = C$

Answer 1

The Correct Option is A

For a curve $xy = C$, consider a point $(a, b)$ on the curve. The equation of the tangent at this point is derived using implicit differentiation: \[ b \, dx + a \, dy = 0 \implies \frac{dy}{dx} = -\frac{b}{a}. \] The equation of the tangent line is: \[ y - b = -\frac{b}{a}(x - a) \implies y = -\frac{b}{a}x + 2b. \] The $x$-intercept is when $y = 0$: \[ 0 = -\frac{b}{a}x + 2b \implies x = 2a. \] The $y$-intercept is when $x = 0$: \[ y = 2b. \] Thus, the intercepts are $(2a, 0)$ and $(0, 2b)$, which are double the coordinates of the point $(a, b)$. Therefore, the family of curves satisfying the given condition is $xy = C$.