Question

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

• A. $(x-1{)}^2=8(y–1)$
• 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

${(y-1)}^2=8(x–1)$

Finding the equation of a parabola.

$\because$The vertex of parabola $(h,k)=(1,1)$

and its focus $(a+h,k)=(3,1)$

$\Rightarrow$$a+h=3$

$\Rightarrow$ $a=2$

$\because$the y-coordinates of vertex of parabola and focus are equal,

$\therefore$the axis of the parabola is parallel to the x-axis.

$\therefore$The equation of the parabola is ${(y–k)}^2=4a(x–h)$

$\Rightarrow$ ${(y–1)}^2=4\times 2(x–1)$

$\Rightarrow$ ${\mathbf{(}\mathbf{y}\mathbf{}\mathbf{–}\mathbf{}\mathbf{1}\mathbf{)}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{8}\mathbf{}\mathbf{(}\mathit{x}\mathbf{}\mathbf{–}\mathbf{}\mathbf{1}\mathbf{)}$

Hence, Option ‘C’ is Correct.