The triangle formed by the tangent to the curve f(x)=x^2+b...

The triangle formed by the tangent to the curve $f(x)=x^{2}+b x-b$ at $(1,1)$ and the co-ordinate axes, lies in the first quadrant. If its area is 2 , then the value of $b$ is
(a) $-1$
(b) 3
(c) $-3$
(d) $\overline{1}$

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Key Concepts

Quadrant Conditions

Quadrant conditions ensure that all points or vertices of a geometric figure lie within a specified region, such as the first quadrant. For the first quadrant, it is required that the x and y coordinates of the intercepts and vertices are positive.

Triangle Area in the Coordinate Plane

When a line intersects the coordinate axes, a right triangle is formed with the intercepts as the legs. The area of this triangle can be calculated as half the product of the absolute values of the x-intercept and y-intercept.

Tangent Line

A tangent line is a straight line that touches a curve at a given point without crossing it at that point. It has the same slope as the curve at that point and is used to approximate the behavior of the curve locally.

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its variable. It is essential in determining the slope of the tangent line at any given point on the curve.

Intercepts of a Line

The x-intercept and y-intercept of a line are the points where the line crosses the x-axis and y-axis, respectively. These intercepts are found by setting the other variable to zero in the linear equation and are key in determining the shape and position of the line in the coordinate plane.

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