Question

find the points on the hyperbola $y^2 - x^2 = 3$ that are closest to the point $(0, 0)$. answer attempt 1 out of 2

Explanation:

Step1: Define distance squared function

The distance from a point $(x,y)$ to $(0,0)$ is $d=\sqrt{x^2+y^2}$. To simplify calculations, minimize the squared distance:
$D(x,y)=x^2+y^2$

Step2: Substitute hyperbola equation

From $y^2 - x^2 = 3$, we get $y^2 = x^2 + 3$. Substitute into $D(x,y)$:
$D(x)=x^2 + (x^2 + 3)=2x^2 + 3$

Step3: Minimize the distance function

$D(x)=2x^2 + 3$ is a quadratic function with a minimum at $x=0$ (since the coefficient of $x^2$ is positive).

Step4: Find corresponding y-values

Substitute $x=0$ into $y^2 - x^2 = 3$:
$y^2 - 0 = 3 \implies y=\pm\sqrt{3}$

Answer:

The closest points are $(0, \sqrt{3})$ and $(0, -\sqrt{3})$