Question

The set of points of the form(t^2+t+1,t^2-t+1),where t is a real number, represents a/an

• A. Circle
• B. Parabola
• C. Ellipse
• D. Hyperbola
• E. pair of straight lines

Answer 1

The Correct Option is B

Given that:

The set of points of the form\( (t^2 + t + 1, t^2 - t + 1)\),

 where \(t\) is a real number, represents a parabola.

Let's analyze the given parametric equations:

\(x = t^2 + t + 1.y = t^2 - t + 1\)

These are parametric equations for the \(x\) and \(y\) coordinates of a point on the plane, where \(t\) is the parameter.

We can eliminate the parameter \(t\) to 

Now can express \(y\) in terms of \(x\), (which will help us identify the geometric shape of the curve.)

Hence we get:

\(x - y = (t^2 + t + 1) - (t^2 - t + 1) \)

\(⇒ 2t x + y = (t^2 + t + 1) + (t^2 - t + 1) \)

                     \(= 2t^2 + 2\)

Now, solving for t in terms of x

 \(t = \dfrac{x - y}{2}\)

Substitute the expression for \(t\) into \(x + y: x + y = 2(\dfrac{x - y}{2})^2 + 2\)

                                                                  ⇒ \(x + y =\dfrac{(x - y)^2}{2} + 2\)

                                                                  ⇒\((x - y)^2 =2x +2y - 4\) 

Now, we have an equation that relates x and y without any parameter t. The equation is a second-degree equation, which represents a parabola.

the vertex of the parabola represented by the equation\( (x - y)^2 = 2x + 2y - 4\) is at the point .\( (1/2, 1/2).\) (Hence it can be finally concluded as a parabola for the given question.)