Question

The parabola having its focus at \[(3,2)\] and directrix along the y-axis has its vertex at
A. \[\left( 2,2 \right)\]
B. \[\left( \dfrac{3}{2},2 \right)\]
C. \[\left( \dfrac{1}{2},2 \right)\]
D. \[\left( \dfrac{2}{3},2 \right)\]

Answer 1

Hint: The focus of parabola is given and the directrix is along y axis that means x is equal to zero. We know the axis of the parabola is perpendicular to its directrix and passes through the focus. The vertex of the parabola will be the midpoint of the foot of the directrix.

Complete step-by-step answer:

Equation of directrix is \[x=0\].

We know the axis of the parabola is perpendicular to its directrix and passes through the focus.

Hence its equation is \[y=2\]

Now the vertex of the parabola will be the midpoint of the foot of the directrix and the focus.

The midpoint of two points, ($x_1$, $y_1$) and ($x_2$, $y_2$) is the point M found by the following formula:

M= \[\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1)

The vertex of the parabola is the midpoint of the directrix and the focus point.

The parabola having its focus at \[(3,2)\], directrix \[\left( 0,2 \right)\]

Hence by substituting the values in (1), we get the vertex of the parabola.

\[\left( \dfrac{3+0}{2},\dfrac{2+2}{2} \right)\] = \[\left( \dfrac{3}{2},2 \right)\]

Therefore the answer is option B.

Note: The vertex of the parabola is the midpoint of the directrix and the focus point and hence the midpoint is found by using the formula. It is given that the directrix is along the y axis so that means the equation x is equal to 0 is the equation of the directrix.