Question

Find a Cartesian equation for the curve. r = 9 sec(e) Identify the curve. hyperbola ellipse circle line O limaçon

Answer 1

The Cartesian equation for the curve r = 9 sec(θ) is x = 9, which represents a vertical line and thus identifies the curve as a line.

Step-by-step explanation:

To find a Cartesian equation for the curve r = 9 sec(θ), we need to convert the given polar equation to Cartesian coordinates (x, y). Recall that in polar coordinates, r is the distance from the origin to a point, and θ is the angle from the positive x-axis to the point.

The conversion is done by using the identities:

• x = r cos(θ)
• y = r sin(θ)

Since sec(θ) = 1/cos(θ), we rewrite r = 9 sec(θ) as r = 9/cos(θ). Multiplying both sides by cos(θ), we get r cos(θ) = 9. Now we use the fact that x = r cos(θ) to substitute x for r cos(θ), yielding the Cartesian equation:

x = 9

This represents a vertical line where every point on the line has an x-coordinate of 9. Therefore, the curve is identified as a line.

Answer 2

To convert the polar equation r = 9 sec(θ) to Cartesian coordinates, we use trigonometric identities and algebra to arrive at x = (x² + y²)/9. Further simplification and completing the square reveals that the equation describes a circle with a radius of 9/2, centered at the point (9/2, 0).

Step-by-step explanation:

To find a Cartesian equation for the given polar equation r = 9 sec(θ), we convert from polar to Cartesian coordinates. In Cartesian coordinates, r is the distance from the origin to the point (x, y), so we have r = √(x² + y²). Secant is the reciprocal of cosine, so sec(θ) = 1/cos(θ). From trigonometry, we know that cos(θ) = x/r. Thus, we can write the equation as r = 9/(x/r), which simplifies to r² = 9x.

Solve for x to get the Cartesian equation: x = r²/9. Substituting r with √(x² + y²), we arrive at x = (x² + y²)/9. Multiplying both sides by 9 gives us 9x = x² + y². Rearranging terms leads to x² - 9x + y² = 0. Completing the square for the x terms gives us x² - 9x + (9/2)² = (9/2)² - y². Therefore, the equation is (x - 9/2)² + y² = (9/2)², which describes a circle of radius 9/2 centered at (9/2, 0).