The equation of the line passing through the point $ (2, 3) $ and making equal intercepts on the coordinate axes is:

Question

If eccentricity of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which passes through the point $(3, 4)$ is $\frac{\sqrt{5}}{3}$, then length of latus rectum of ellipse, is :

Answer 1

Step 1: Formulate the general equation for a line with equal intercepts. A line with identical intercepts $ a $ on the x- and y-axes is represented by: \[ \frac{x}{a} + \frac{y}{a} = 1 \Rightarrow \frac{x + y}{a} = 1 \Rightarrow x + y = a \] Step 2: Determine the intercept $ a $ using the provided point. Substitute the point $ (2, 3) $ into the equation $ x + y = a $: \[ 2 + 3 = a \Rightarrow a = 5 \Rightarrow \text{The line's equation is } x + y = 5 \] Step 3: Confirm the line's passage through (2, 3). \[ x + y = 2 + 3 = 5 \Rightarrow \text{Verification successful} \]

The equation of the line passing through the point $ (2, 3) $ and making equal intercepts on the coordinate axes is:

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The Correct Option is A

Solution and Explanation

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