Find the equation of parabola whose latus rectum is the line segment joining the points (-3, 1), (1, 1)

Friday, April 27, 2012

Find the equation of parabola whose latus rectum is the line segment joining the points (-3, 1), (1, 1)

You need to remember the equation of parabola, such that:

$\displaystyle{4}{p}{\left({x}-{h}\right)}={{\left({y}-{k}\right)}}^{{2}}$

The length of latus rectum is |4p| and it must be set equal to the length of the segment joining the points (-3,1), (1,1).

You need to find the length of the segment joining the points (-3,1), (1,1), using the distance formula, such that:

$\displaystyle{d}=\sqrt{{{{\left(-{3}-{1}\right)}}^{{2}}+{{\left({1}-{1}\right)}}^{{2}}}}$

$\displaystyle{d}=\sqrt{{16}}$

$\displaystyle{d}={4}$

Hence, the length of latus rectum is:

$\displaystyle{\left|{4}{p}\right|}={4}\Rightarrow{p}=\pm{1}$

The focus of parabola is the midpoint of the segment line joining the points (-3,1), (1,1), hence, you may evaluate the coordinates of the midpoint, such that:

$\displaystyle{x}=\frac{{-{3}+{1}}}{{2}}\Rightarrow{x}=-{1}$

$\displaystyle{y}=\frac{{{1}+{1}}}{{2}}\Rightarrow{y}={1}$

The coordinates of the focus are (-1,1).

The vertex of parabola is $\displaystyle{\left(-{1}\pm{p},{1}\right)}$ , hence, the coordinates of the vertex are $\displaystyle{\left(-{1}\pm{1},{1}\right)}.$

The parabola that opens to the right has the length of latus rectum $\displaystyle{4}{p}={4}$ , the vertex $\displaystyle{\left({h},{k}\right)}={\left(-{1}-{1},{1}\right)}={\left(-{2},{1}\right)}$ and the equation of parabola $\displaystyle{4}{\left({x}+{2}\right)}={{\left({y}-{1}\right)}}^{{2}}.$

The parabola that opens to the left has the length of latus rectum $\displaystyle{4}{p}=-{4}$ , the vertex $\displaystyle{\left({h},{k}\right)}={\left(-{1}+{1},{1}\right)}={\left({0},{1}\right)}$ and the equation of parabola $\displaystyle-{4}{\left({x}-{0}\right)}={{\left({y}-{1}\right)}}^{{2}}.$

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Friday, April 27, 2012

Find the equation of parabola whose latus rectum is the line segment joining the points (-3, 1), (1, 1)

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