Chapter 10: Q 64. (page 655)

Find an equation for each ellipse. Graph the equation by hand.
Center at $(1, 2)$ vertex at $(1, 4)$ contains the point $(1 + \sqrt{3}, 3)$ .

Short Answer

Expert verified

The equation of the ellipse is $\frac{{(x - 1)}^{2}}{4} + \frac{{(y - 2)}^{2}}{4} = 1$ and graph of the ellipse is

Step by step solution

. Given information.

Center at $(1, 2)$ vertex at $(1, 4)$ contains the points $(1 + \sqrt{3}, 3)$ .

On comparing with the standard equation of the ellipse $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$

Here, $x = 1 + \sqrt{3}, y = 3, h = 1, k = 2$ and $k + a = 4 a = 4 - 2 a = 2$

Step 2. The equation of ellipse.

Substituting the values in the equation of ellipse.

$\frac{{(1 + \sqrt{3} - 1)}^{2}}{2^{2}} + \frac{{(3 - 2)}^{2}}{b^{2}} = 1 \frac{{(\sqrt{3})}^{2}}{4} + \frac{1}{b^{2}} = 1 \frac{1}{b^{2}} = 1 - \frac{3}{4} \frac{1}{b^{2}} = \frac{1}{4} b^{2} = 4$

Thus, the equation of the ellipse is $\frac{{(x - 1)}^{2}}{4} + \frac{{(y - 2)}^{2}}{4} = 1$

Step 3. Graph of the ellipse .

The major axis is parallel to the y-axis. so the vertices are $a = 2$ units up and down of the center $(1, 2)$ . Therefore the vertices are $v_{1} = (1, 0)$ and $v_{2} = (1, 4)$ .

Since,

c^{2} = a^{2} - b^{2} c^{2} = 4 - 4 c^{2} = 0 c = 0

The major axis is parallel to the y-axis foci are $(1, 2 - 0)$ and $(1, 2 + 0)$ or $(1, 2)$ .

Finally, we use the value $b = 2, b = 3$ to find the two points above and below the center as $(- 1, 2)$ and $(3, 2)$ .

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Find an equation for each ellipse. Graph the equation by hand.
Center at $(1, 2)$ vertex at $(1, 4)$ contains the point $(1 + \sqrt{3}, 3)$ .

Short Answer

Step by step solution

. Given information.

Step 2. The equation of ellipse.

Step 3. Graph of the ellipse .

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Find an equation for each ellipse. Graph the equation by hand. Center at(1,2)vertex at(1,4) contains the point(1+3,3).

Short Answer

Step by step solution

. Given information.

Step 2. The equation of ellipse.

Step 3. Graph of the ellipse .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Statistics

Applied Mathematics

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.

Find an equation for each ellipse. Graph the equation by hand.
Center at $(1, 2)$ vertex at $(1, 4)$ contains the point $(1 + \sqrt{3}, 3)$ .