Question

The equation of the parabola with focus \((2,0)\) and vertex \((1,0)\) is

• A. \(y^2=4x\)
• B. \(y^2=4x-4\)
• C. \(y^2=4(x+1)\)
• D. \(y^2=-4(x-1)\)

Hint:

For a parabola opening along positive \(x\)-axis, use \((y-k)^2=4a(x-h)\), where vertex is \((h,k)\) and focus is \((h+a,k)\).

Answer 1

The Correct Option is B

We are given the vertex of the parabola: \[ (1,0). \] The focus is: \[ (2,0). \] Since both vertex and focus lie on the \(x\)-axis, the axis of the parabola is horizontal. The standard equation of a parabola with vertex \((h,k)\) and opening along positive \(x\)-axis is: \[ (y-k)^2=4a(x-h). \] Here, \[ (h,k)=(1,0). \] So, \[ h=1,\qquad k=0. \] The focus of this parabola is \[ (h+a,k). \] Given focus is \[ (2,0). \] Therefore, \[ h+a=2. \] Since \(h=1\), \[ 1+a=2. \] \[ a=1. \] Now substitute \(h=1\), \(k=0\), and \(a=1\) in the standard equation: \[ (y-0)^2=4(1)(x-1). \] \[ y^2=4(x-1). \] \[ y^2=4x-4. \] Hence, the equation of the parabola is \[ y^2=4x-4. \]