Find the co-ordinates of that point on the curve `x^(2)/a^(2)+y^(2)/b^(2) = 1` at which the tangent drawn is parallel to Y-axis.

Answer

To find the coordinates of the point on the curve \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where the tangent is parallel to the Y-axis, we can follow these steps:

### Step 1: Understand the condition for the tangent to be parallel to the Y-axis
A tangent that is parallel to the Y-axis means that the slope of the tangent line is undefined, which occurs when the derivative \( \frac{dy}{dx} \) is infinite. This typically happens when \( \frac{dx}{dy} = 0 \).

### Step 2: Differentiate the curve implicitly
Starting from the equation of the curve:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
we differentiate both sides with respect to \( x \):
\[ \frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0 \]
Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{2y}{b^2} \frac{dy}{dx} = -\frac{2x}{a^2} \]
\[ \frac{dy}{dx} = -\frac{b^2 x}{a^2 y} \]

### Step 3: Set the condition for the tangent to be vertical
For the tangent to be vertical, we need \( \frac{dy}{dx} \) to be undefined, which occurs when \( y = 0 \) (since the denominator becomes zero).

### Step 4: Substitute \( y = 0 \) into the curve equation
Substituting \( y = 0 \) into the original curve equation:
\[ \frac{x^2}{a^2} + \frac{0^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{x^2}{a^2} = 1 \]
Multiplying both sides by \( a^2 \):
\[ x^2 = a^2 \]

### Step 5: Solve for \( x \)
Taking the square root of both sides gives:
\[ x = a \quad \text{or} \quad x = -a \]

### Step 6: Write the coordinates of the points
Thus, the coordinates of the points where the tangent is parallel to the Y-axis are:
\[ (a, 0) \quad \text{and} \quad (-a, 0) \]

### Final Answer
The coordinates of the points are \( (a, 0) \) and \( (-a, 0) \).

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