Question

Equation of the parabola with its vertex at $(1,1)$ and focus $(3,1)$ is

• A. $(x1{)}^2=8(y-1)$
• B. $(y-1{)}^2=8(x-3)$
• C. $(y-1{)}^2=8(x-1)$
• D. $(x-3{)}^2=8(y-1)$

Answer 1

The correct option is C $(y-1{)}^2=8(x-1)$

As the vertex is$(1,1)$ and focus is $(3,1)$, whose ordinate is same its axis of symmetry is $y=1$.

And as vertex is equidistant from foci and directrix, and latter is perpendicular to axis of symmetry.

Directrix is $x=1$

As parabola is the locus of a point whose distance from directrix $x+1=0$ and focus $(3,1)$

Its equation is $(x-3{)}^2+(y-1{)}^2=(x+1{)}^2$

$\Rightarrow x^2-6X+9+y^2-2Y+1=x^2+2X+1$

$\Rightarrow y^2-2y+9=8x$

$\Rightarrow (y-1{)}^2=8(x-1)$

So, option C is the answer.