Question

Equation of parabola with its vertex at \[(1,1)\] and focus \[(3,1)\] is
1) \[{(x - 1)^2} = 8(y - 1)\]
2) \[{(y - 1)^2} = 8(x - 3)\]
3) \[{(y - 1)^2} = 8(x - 1)\]
4) \[{(x - 3)^2} = 8(y - 1)\]

Answer 1

Hint:since vertex and focus of the parabola is given to us so we can note that the directrix of the parabola is x=1 and as both vertex and Focus is in the first quadrant so we can find the equation of the parabola by considering standard form of the parabola.

Complete step by step answer:
As the Vertex is \[(1,1)\] and focus \[(3,1)\] From these coordinates we can note that the value of ordinate is the same therefore, we can say that the axis of symmetry is \[y = 1\]. And as vertex is equidistant from foci and directrix, whose distance from directrix \[x + 1 = 0\] and focus \[(3,1)\] so the equation for this parabola will be of the form
\[{\left( {x - 3} \right)^2} + {\left( {y - 1} \right)^2} = {\left( {x + 1} \right)^2}\]
On expanding the above equation, we get
\[{x^2} - 6x + 9 + {y^2} - 2y + 1 = {x^2} + 2x + 1\]
On simplification we get
\[{y^2} + 9 - 2y = 2x + 1 + 6x - 1\]
\[{y^2} + 9 - 2y = 2x + 6x\]
To arrange it in certain form as per the given options let us add and subtract 1 to the LHS so we get
\[{y^2} + 9 - 2y + 1 - 1 = 2x + 6x\]
Let us rearrange the above equation we get
\[{y^2} - 2y + 1 = 8x - 8\]
\[ \Rightarrow {(y - 1)^2} = 8(x - 1)\]is the required equation of parabola.

So, the correct answer is “Option 3”.

Note: Parabolas are commonly known as the graphs of quadratic functions. That is every quadratic equation represents a parabola. A parabola is a set of all points or locus of points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola and the line is called the directrix. The focus lies on the axis of symmetry of the parabola.