Question

The eccentricity of the ellipse, which meets the straight line \( \frac{x}{7} + \frac{y}{2} = 1 \) on the axis of \( x \) and the straight line \( \frac{x}{3} - \frac{y}{5} = 1 \) on the axis of \( y \), and whose axes lie along the axes of coordinates, is:

• A. \( \frac{3\sqrt{2}}{7} \)
• B. \( \frac{2\sqrt{6}}{7} \)
• C. \( \frac{\sqrt{3}}{7} \)
• D. None of these

Hint:

To find the eccentricity of an ellipse, use the formula \( e = \sqrt{1 - \frac{b^2}{a^2}} \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

Answer 1

The Correct Option is B

We are given the following equations of two straight lines: 1. The line \( \frac{x}{7} + \frac{y}{2} = 1 \), which intersects the \( x \)-axis and \( y \)-axis. 2. The line \( \frac{x}{3} - \frac{y}{5} = 1 \), which also intersects the \( x \)-axis and \( y \)-axis. These two lines are the axes of the ellipse. To find the eccentricity of the ellipse, we first need to identify the semi-major axis \( a \) and semi-minor axis \( b \) from the intersection points of the ellipse with the coordinate axes. Step 1: Finding the intercepts From the first line, \( \frac{x}{7} + \frac{y}{2} = 1 \), the intercepts on the axes are: - When \( y = 0 \), \( \frac{x}{7} = 1 \) gives \( x = 7 \), so the \( x \)-intercept is \( 7 \). - When \( x = 0 \), \( \frac{y}{2} = 1 \) gives \( y = 2 \), so the \( y \)-intercept is \( 2 \). From the second line, \( \frac{x}{3} - \frac{y}{5} = 1 \), the intercepts are: - When \( y = 0 \), \( \frac{x}{3} = 1 \) gives \( x = 3 \), so the \( x \)-intercept is \( 3 \). - When \( x = 0 \), \( -\frac{y}{5} = 1 \) gives \( y = -5 \), so the \( y \)-intercept is \( -5 \). Step 2: Identifying semi-major axis \( a \) and semi-minor axis \( b \) The equation of the ellipse will have the form: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \] The semi-major axis \( a \) corresponds to the larger of the intercepts, and the semi-minor axis \( b \) corresponds to the smaller intercept. From the intercepts, the semi-major axis is \( a = 7 \) (since the \( x \)-intercept of 7 is the largest), and the semi-minor axis is \( b = 5 \) (since the \( y \)-intercept of 5 is the largest in magnitude). Step 3: Calculating the eccentricity The eccentricity \( e \) of an ellipse is given by: \[ e = \sqrt{1 - \frac{b^2}{a^2}}. \] Substitute \( a = 7 \) and \( b = 5 \): \[ e = \sqrt{1 - \frac{5^2}{7^2}} = \sqrt{1 - \frac{25}{49}} = \sqrt{\frac{49}{49} - \frac{25}{49}} = \sqrt{\frac{24}{49}} = \frac{\sqrt{24}}{7} = \frac{2\sqrt{6}}{7}. \] Thus, the eccentricity of the ellipse is \( \frac{2\sqrt{6}}{7} \), and the correct answer is (b).