Geo5.1.40

From Example Problems
Jump to: navigation, search

Prove that the locus of midpoints of chords of constant length 2l of the parabola y^{2}=4ax\, is (y^{2}-4ax)(y^{2}+4a^{2})+4a^{2}l^{2}=0\,

Let P(at_{1}^{{2}},2at_{1}),Q(at_{2}^{{2}},2at_{2})\, be the two extremeties of the given

parabola sothat PQ=2l. let this be equation 1.

Let R(x_{1},y_{1})\, be the midpoint of the chord PQ.Then

x_{1}={\frac  {at_{1}^{{2}}+at_{2}^{{2}}}{2}}={\frac  {a(t_{1}^{{2}}+t_{2}^{{2}})}{2}},y_{1}=a(t_{1}+t_{2})\,

From the above two equalities,we have

y_{1}^{{2}}=a^{2}(t_{1}^{{2}}+t_{2}^{{2}}+2t_{1}t_{2})\,

y_{1}^{{2}}=a^{2}({\frac  {2x_{1}}{a}}+2t_{1}t_{2})\,

t_{1}t_{2}={\frac  {y_{1}^{{2}}-2ax_{1}}{2a^{2}}}\,

From equation 1,

4l^{2}=PQ^{{2}}=a^{2}[t_{1}^{{2}}-t_{2}^{{2}}]^{2}+4a^{2}[t_{1}-t_{2}]^{2}\,

2l^{2}=a^{2}(t_{1}-t_{2})^{2}[(t_{1}+t_{2})^{2}+4]=a^{2}[(t_{1}+t_{2})^{2}-4t_{1}t_{2}][(t_{1}+t_{2})^{2}+4]\,

4a^{2}l^{2}=(y_{1}^{{2}}-2y_{1}^{{2}}+4ax_{1})(y_{1}^{{2}}+4a^{2})\,

Therefore,the locus of (x_{1},y_{1})\, is

4a^{2}l^{2}+(y^{2}-4ax)(y^{2}+4a^{2})=0\,

Main Page:Geometry:The Parabola