Question

При каком p вершина параболы y = x^ 2 + px + 58 находится на расстоянии 10 от начала координат, если известно, что вершина параболы лежит в третьей четверти.

Answer 1

1) Координаты вершины параболы - точка (x', y'):
$x'=- \frac{p}{2} ;\ y'=y(- \frac{p}{2})= \frac{p^2}{4}- \frac{p^2}{2}+58=58-\frac{p^2}{4}\\ \Rightarrow (- \frac{p}{2};\ 58-\frac{p^2}{4})$
2) Т.к. вершина параболы - в III четверти, то x'<0 и y'<0, т.е.
$\begin {cases} - \frac{p}{2} \ \textless \ 0 \\ 58-\frac{p^2}{4}\ \textless \ 0 \end {cases} \ \Leftrightarrow \begin {cases} p\ \textgreater \ 0 \\ p^2}\ \textgreater \ 232 \end {cases} \ \Rightarrow p\ \textgreater \ \sqrt{232}$
3) Расстояние до начала координат равно 10 и задается уравнением:
$(- \frac{p}{2})^2+(58- \frac{p^2}{4} )^2=10^2 \\ \frac{p^2}{4}+(58- \frac{p^2}{4} )^2=100$
Замена $t=\frac{p^2}{4},\ t\ \textgreater \ 58$
t + (58 - t)² - 100 = 0
t² - 115t + 3264 = 0
D = 13225-13026 = 169
t=51 или t=64
Требованию t>58 удовлетворяет t=64, поэтому
$\frac{p^2}{4}=64 \Rightarrow p = \pm 16$
Т.к. $p\ \textgreater \ \sqrt{232}$, то р=16.
Ответ: р=16.

Answer 2

Finding the Vertex of a Parabola

To find the vertex of the parabola given by the equation y = x^2 + px + 58 and determine the distance from the origin, we can use the fact that the vertex of a parabola in the form y = ax^2 + bx + c is given by (-b/2a, c - b^2/4a). Since the vertex lies in the third quadrant, both the x and y coordinates of the vertex will be negative.

Calculating the Vertex Coordinates

Using the formula for the x-coordinate of the vertex, we have x = -p/2. Substituting this into the equation for the parabola, we can solve for the y-coordinate of the vertex.

The distance from the origin to the vertex can be calculated using the distance formula d = sqrt(x^2 + y^2).

Solution

Given the equation y = x^2 + px + 58 and the fact that the vertex lies in the third quadrant, we can calculate the coordinates of the vertex and the distance from the origin as follows:

1. The x-coordinate of the vertex is given by x = -p/2. 2. Substituting this into the equation for the parabola, we get y = (-p/2)^2 + p*(-p/2) + 58. 3. Simplifying, we find y = p^2/4 - p^2/2 + 58. 4. The y-coordinate of the vertex is y = p^2/4 - p^2/2 + 58. 5. The distance from the origin to the vertex is given by d = sqrt((-p/2)^2 + (p^2/4 - p^2/2 + 58)^2).

Conclusion

By solving for the vertex coordinates and the distance from the origin, we can determine the position of the vertex of the parabola y = x^2 + px + 58 in the third quadrant.