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Chapter 6: Problem 13

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (卤7,0)\(;\) foci: (卤2,0)

Short Answer

Expert verified
The standard form of the equation of the ellipse with given vertices and foci is \(\frac{x^2}{49} + \frac{y^2}{45} = 1\)

Step by step solution

01

Determine the semi-major axis

The vertices of the ellipse give the length of the major axis. In this case, the vertices are at (卤7,0), thus, the length of the major axis is 2*7 = 14. Therefore, the semi-major axis 'a' will be 7.
02

Determine the distance from the center to the Foci

The foci of the ellipse are given as (卤2,0), thus, the distance from the center to the foci, 'c', is 2.
03

Calculate the semi-minor axis

Using the relation \(a^2 = b^2 + c^2\) , for the given ellipse, we can find out the value of 'b'. In this relation, 'a' is the semi-major axis, 'b' is the semi-minor axis and 'c' is the distance from the center to a foci. Here, 'a' is 7 and 'c' is 2. So, \(b^2 = a^2 - c^2 = 7^2 - 2^2 = 49-4 = 45\). Hence, 'b' is \(\sqrt{45}\).
04

Write the standard form of the ellipse

Now we have the values of 'a' and 'b', the standard form of the ellipse is found by replacing 'a' and 'b' in the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). So, the equation of the ellipse is \(\frac{x^2}{7^2} + \frac{y^2}{{\sqrt{45}}^2} = 1\) or \(\frac{x^2}{49} + \frac{y^2}{45} = 1\).

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