Respuesta :
Answer:
The correct option is option (C).
The required equation of the parabola is
[tex](x-2)^2=8(y-1)[/tex]
Step-by-step explanation:
Formula:
- The distance of a point (x₁,y₁) from a line ax+by+c=0 is
[tex]\frac{ax_1+by_1+c}{\sqrt{a^2+b^2}}[/tex].
- The distance between two points (x₁,y₁) and (x₂,y₂) is [tex]\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Given that, the focus of the parabola is (2,3) and directrix y= - 1.
We know that, a parabola is the locus of points that equidistant from the directrix and the focus.
Let any point on the parabola be P(x,y) .
The distance of P from the directrix is
[tex]=\frac{y+1}{\sqrt {1^2}}[/tex]
=y+1
The distance of the point P from the focus (2,3) is
[tex]\sqrt{(x-2)^2+(y-3)^2}[/tex]
According to the problem,
[tex]\sqrt{(x-2)^2+(y-3)^2}=y+1[/tex]
[tex]\Rightarrow (x-2)^2+(y-3)^2=(y+1)^2[/tex]
[tex]\Rightarrow (x-2)^2+y^2-6y+9=y^2+2y+1[/tex]
[tex]\Rightarrow (x-2)^2=y^2+2y+1-y^2+6y-9[/tex]
[tex]\Rightarrow (x-2)^2=8y-8[/tex]
[tex]\Rightarrow (x-2)^2=8(y-1)[/tex]
The required equation of the parabola is
[tex](x-2)^2=8(y-1)[/tex]
