The Tangent at point P of a curve meets the y axis at B,the line through P parallel to y axis meets the x axis at A. If the area of AOB traingle is constant, the curve is a

a)parabola

b)hyperbola

c)ellipse

d)circle

To determine the type of curve given the conditions of the problem, let’s analyze the information step by step. We have a curve with a tangent at point P that intersects the y-axis at point B. Additionally, a line through P that is parallel to the y-axis meets the x-axis at point A. The area of triangle AOB remains constant as P moves along the curve. This scenario provides significant insight into the nature of the curve.

Understanding the Triangle AOB

The triangle AOB consists of points A, O (the origin), and B. The area of a triangle formed by points on a coordinate plane can be calculated using the formula:

• Area = (1/2) * base * height

In this context, the base can be considered as the distance from O to A along the x-axis, while the height is the distance from O to B along the y-axis. Since the area of triangle AOB is constant, it implies that the product of the base and height must remain constant as point P moves along the curve.

Implications for the Curve

Given that the area remains constant, we can express this condition mathematically. Let’s denote the coordinates of points A and B as follows:

• A = (x_A, 0)
• B = (0, y_B)

The area can therefore be represented as:

• Area = (1/2) * |x_A| * |y_B| = k, where k is a constant.

Exploring the Relationship

From the area equation, we can rearrange it to find a relationship between x_A and y_B:

• |y_B| = (2k) / |x_A|.

This expression indicates that as the x-coordinate of point A changes, the y-coordinate of point B must adjust in a way that their product remains constant. Such a relationship is characteristic of a hyperbola, where the product of the distances from the axes to the curve defines its shape.

Identifying the Curve Type

To further confirm that the curve is indeed a hyperbola, we consider the general equation of a hyperbola centered at the origin:

• (x^2/a^2) - (y^2/b^2) = 1

In this equation, the distances from the center to the vertices along the x-axis and y-axis are inversely related, which aligns perfectly with our derived relationship from the triangle’s area being constant.

Conclusion on Curve Type

Given that the area of triangle AOB remains constant as point P moves along the curve, and considering the mathematical implications of the relationships we established, we conclude that the curve in question is a hyperbola. Therefore, the correct answer to your question is option b) hyperbola.