If the eccentricities of the two ellipse `(x^(2))/(169)+(y^(2))/(25)=1 and (x^(2))/(a^(2))+(y^(2))/(b^(2))=1` and equal , then the value `(a)/(b) ` , is

Options

A. `5/13`
B. `6/13`
C. `13/5`
D. `13/6`

Answer

To solve the problem, we need to find the value of \( \frac{a}{b} \) given that the eccentricities of two ellipses are equal. Let's break it down step by step.

### Step 1: Identify the eccentricity of the first ellipse
The first ellipse is given by the equation:
\[ \frac{x^2}{169} + \frac{y^2}{25} = 1 \]
Here, \( a^2 = 169 \) and \( b^2 = 25 \). The eccentricity \( e_1 \) of an ellipse is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values:
\[ e_1 = \sqrt{1 - \frac{25}{169}} \]

### Step 2: Simplify the expression for \( e_1 \)
Calculating \( \frac{25}{169} \):
\[ e_1 = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{169 - 25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13} \]

### Step 3: Identify the eccentricity of the second ellipse
The second ellipse is given by the equation:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
The eccentricity \( e_2 \) of this ellipse is given by:
\[ e_2 = \sqrt{1 - \frac{b^2}{a^2}} \]

### Step 4: Set the eccentricities equal
According to the problem, the eccentricities of the two ellipses are equal:
\[ e_1 = e_2 \]
Thus, we have:
\[ \frac{12}{13} = \sqrt{1 - \frac{b^2}{a^2}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{12}{13}\right)^2 = 1 - \frac{b^2}{a^2} \]
Calculating \( \left(\frac{12}{13}\right)^2 \):
\[ \frac{144}{169} = 1 - \frac{b^2}{a^2} \]

### Step 6: Rearranging the equation
Rearranging the equation gives:
\[ \frac{b^2}{a^2} = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169} \]

### Step 7: Take the square root
Taking the square root of both sides:
\[ \frac{b}{a} = \frac{5}{13} \]

### Step 8: Find \( \frac{a}{b} \)
To find \( \frac{a}{b} \), we take the reciprocal:
\[ \frac{a}{b} = \frac{13}{5} \]

### Final Answer
Thus, the value of \( \frac{a}{b} \) is:
\[ \frac{a}{b} = \frac{13}{5} \]