Question

Написать уравнение эллипса у которого сумма полуосей равна 36 а расстояние между фокусами лежащими на оси 0x равно 48

Answer 1

Дано: эллипс, а+в = 36, 2с = 48.
Уравнение эллипса $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1.$
Из уравнения с² = а² - в² = (а - в)(а + в) находим :
а - в = с²/(а + в) = 24²/36 =  576/36  = 16 .
Составим систему:
$\left \{ {{a+b=36} \atop {a-b=16}} \right.$.
Сложим уравнения: 2а = 52, отсюда находим полуось а = 52/2 = 26.
Вторая полуось в = 36 - а = 36 - 26 = 10.
Составим уравнение эллипса: 
$\frac{x^2}{26^2} + \frac{y^2}{10^2}=1.$

Answer 2

Writing the Equation of an Ellipse

To write the equation of an ellipse, we can use the standard form of the equation for an ellipse with its center at the origin. The equation is given by:

x^2/a^2 + y^2/b^2 = 1

Where: - a is the distance from the center to the vertices along the x-axis. - b is the distance from the center to the vertices along the y-axis.

Using the Given Information to Write the Equation

We are given that the sum of the semi-axes is 36 and the distance between the foci on the x-axis is 48.

The relationship between the sum of the semi-axes and the distance between the foci for an ellipse is given by the equation:

2a = 36 2c = 48

Where: - a is the distance from the center to the vertices along the x-axis. - c is the distance from the center to each focus along the x-axis.

Solving for a and c

From the given information, we can solve for a and c:

a = 36 / 2 = 18 c = 48 / 2 = 24

Writing the Final Equation

Now that we have the values of a and c, we can write the equation of the ellipse:

x^2/18^2 + y^2/b^2 = 1

Where: - a = 18 - c = 24

This is the equation of the ellipse with the given properties.