Question

Расстояние от вершины параболы y=(x+n)^2+3 до начала координат равно 5, если n равно? С решением

Answer 1

Тут всё ясно. Координаты вершины уже заложены в уравнении параболы. Это (-n;3) (т.е. параболу y=x² сместили на 3 ед. вверх по у и на -n единиц по x). Расстояние между (-n;3) и (0;0) : $\sqrt{(0-(-n))^2+(0-3)^2}=5; \sqrt{n^2+9}=5; n^2+9=25;$
$n^2=16;$, здесь подходят значения 4 и -4. Оно и понятно, ведь можно найти такую параболу слева и справа.

Answer 2

Solution for Finding the Value of n in the Parabola Equation

To find the value of n in the parabola equation y = (x + n)^2 + 3 when the distance from the vertex to the origin is 5, we can use the distance formula to calculate the distance between two points.

The distance formula between two points (x1, y1) and (x2, y2) is given by: √((x2 - x1)^2 + (y2 - y1)^2)

In this case, the vertex of the parabola is at the point (-n, 3), and the origin is at the point (0, 0). We can use the distance formula to find the value of n when the distance is 5.

Calculating the Distance

Using the distance formula, the distance between the vertex (-n, 3) and the origin (0, 0) is: √((0 - (-n))^2 + (0 - 3)^2) = 5

Solving for n: √(n^2 + 9) = 5

Squaring both sides to eliminate the square root: n^2 + 9 = 25

Subtracting 9 from both sides: n^2 = 16

Taking the square root of both sides: n = ±4

Conclusion

The value of n in the parabola equation y = (x + n)^2 + 3 when the distance from the vertex to the origin is 5 can be either n = 4 or n = -4.