Question

The coordinates of end point of the latus rectum of the parabola ${\left( {{\text{y - 1}}} \right)^2} = 4\left( {{\text{x + 1}}} \right)$are
${\text{A}}{\text{.}}\left( {0, - 3} \right) \\\ {\text{B}}{\text{.}}\left( {0, - 1} \right) \\\ {\text{C}}{\text{.}}\left( {0,1} \right) \\\ {\text{D}}{\text{.}}\left( {1,3} \right) \\\$

Answer 1

Hint: Since, this is a question of parabola with shifted vertex. We first need to find the vertex of the parabola and then find the values of the end points of the latus rectum. For this we also need to understand the concept of latus rectum and then choose the correct option among the given options.

Complete step-by-step answer:

The given equation of parabola ${\left( {{\text{y - 1}}} \right)^2} = 4\left( {{\text{x + 1}}} \right)$ can be rewritten as ${{\text{Y}}^2} = 4{\text{X}}$….(1)
Where, ${\text{Y = y - 1 and X = x + 1}}$
$\Rightarrow {\text{y = Y + 1 and x = X - 1}}$
Thus, it is shifting the origin at $\left( {1, - 1} \right)$
$\therefore {\text{From Eq}}{\text{.(1)}} \\\ {{\text{Y}}^2} = 4{\text{X}} \\\$
The coordinates of the endpoints of latus rectum are
${\text{X = 1,Y = 2 and X = 1,Y = - 2 }}$
$\therefore {\text{ x = X - 1 and y = Y + 1}}$
So, the coordinates of x and y are (0,3) and (0,-1) respectively.

Therefore we can find the coordinates of the endpoints of the latus rectum by finding the values of ${\text{x,y}}$ which are (0,3) and (0,-1) . Since we got two answers we have to check the options and mark the correct answer according to the options.

Note: The concept of latus rectum is very important to solve the given question and we should clearly know how to solve the parabola which has its origin at a different point than (0,0) . As we know, the latus rectum is the focal chord at right angles to the axis of the parabola. Thus, by simplifying this equation we can get our answer.