Calculating Eccentricity and Semi-Latus Rectum from Polar Equation

The given equation of the curve in polar coordinates is:

\(\frac{1}{r} = 8 + 5 \cos \theta\)

This equation represents a conic section. The standard form of the polar equation for a conic section with the focus at the origin is:

\(\frac{l}{r} = 1 + e \cos \theta\)

where:

• \(l\) is the semi-latus rectum
• \(e\) is the eccentricity

To find the eccentricity and semi-latus rectum from the given equation, we need to transform it into the standard form. The standard form requires the constant term on the right side to be 1.

Divide the given equation by 8:

\(\frac{1}{r} \times \frac{1}{8} = \frac{8}{8} + \frac{5}{8} \cos \theta\)

This simplifies to:

\(\frac{1}{8r} = 1 + \frac{5}{8} \cos \theta\)

Now, we can rewrite the left side to match the standard form \(\frac{l}{r}\). We can express \(\frac{1}{8r}\) as \(\frac{1/8}{r}\).

\(\frac{1/8}{r} = 1 + \frac{5}{8} \cos \theta\)

Comparing this equation with the standard form \(\frac{l}{r} = 1 + e \cos \theta\), we can identify the values of \(l\) and \(e\).

• The eccentricity \(e\) is the coefficient of \(\cos \theta\), which is \(\frac{5}{8}\).
• The semi-latus rectum \(l\) is the constant term in the numerator on the left side, which is \(\frac{1}{8}\).

So, the eccentricity \(e = \frac{5}{8}\) and the semi-latus rectum \(l = \frac{1}{8}\).

The question asks for the eccentricity and semi-latus rectum respectively. Therefore, the answer is \(\frac{5}{8}\) and \(\frac{1}{8}\).

Let's look at the options provided:

• Option 1: \(\frac{5}{8}\), 8
• Option 2: \(\frac{1}{8}\), 8
• Option 3: \(\frac{1}{8}\), \(\frac{5}{8}\)
• Option 4: \(\frac{5}{8}\), \(\frac{1}{8}\)

Comparing our calculated values with the options, we find that option 4 matches our results.

The eccentricity is \(\frac{5}{8}\).

The semi-latus rectum is \(\frac{1}{8}\).