Question

Given a curve C. Let the tangent line at P(x, y) on C is perpendicular to the line joining P and Q(1, 0). If the line 2x + 3y - 15 = 0 is tangent to the curve C, then the length of the tangent from the point (5, 0) to the curve C is $\sqrt{n}$ (where n $\in$ M). Find the value of n

Answer 1

3

Answer 2

The first condition defines a differential equation $y \frac{dy}{dx} = 1-x$, which upon integration yields the equation of a circle $(x-1)^2 + y^2 = R^2$ centered at (1,0). The second condition, that the line $2x + 3y - 15 = 0$ is tangent to this circle, allows us to calculate the radius $R$ by finding the distance from the center (1,0) to the line, yielding $R = \sqrt{13}$. Thus, the curve C is the circle $(x-1)^2 + y^2 = 13$. The length of the tangent from an external point $(x_0, y_0)$ to a circle $(x-h)^2 + (y-k)^2 = R^2$ is given by $\sqrt{(x_0-h)^2 + (y_0-k)^2 - R^2}$. For the point (5,0) and the circle $(x-1)^2 + y^2 = 13$, this length is $\sqrt{(5-1)^2 + (0-0)^2 - 13} = \sqrt{16 - 13} = \sqrt{3}$. Given this length is $\sqrt{n}$, we find $n=3$.