Which of the following is a point at which the ellipse `(x^(2))/(9)+(y^(2))/(16)=1` intersects the y - axis ?

Options

A. (0, -3)
B. (0, -4)
C. (0, -8)
D. (0, -9)

Explanation

To find the points at which the ellipse \(\frac{x^2}{9} + \frac{y^2}{16} = 1\) intersects the y-axis, we can follow these steps:

### Step 1: Understand the Intersection with the Y-axis
The points on the y-axis have an x-coordinate of 0. Therefore, we can denote the point of intersection as \((0, y_1)\).

### Step 2: Substitute x = 0 into the Ellipse Equation
Substituting \(x = 0\) into the equation of the ellipse:
\[ \frac{0^2}{9} + \frac{y_1^2}{16} = 1 \]
This simplifies to:
\[ 0 + \frac{y_1^2}{16} = 1 \]
Thus, we have:
\[ \frac{y_1^2}{16} = 1 \]

### Step 3: Solve for \(y_1^2\)
To isolate \(y_1^2\), multiply both sides by 16:
\[ y_1^2 = 16 \]

### Step 4: Take the Square Root
Taking the square root of both sides gives us:
\[ y_1 = \pm 4 \]
This means:
\[ y_1 = 4 \quad \text{or} \quad y_1 = -4 \]

### Step 5: Write the Points of Intersection
The points at which the ellipse intersects the y-axis are:
\[ (0, 4) \quad \text{and} \quad (0, -4) \]

### Step 6: Identify the Correct Option
From the given options, we can see that \( (0, -4) \) is one of the points of intersection.

### Final Answer
The point at which the ellipse intersects the y-axis is \((0, -4)\).

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