Geo5.1.15

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The line 2x-y+2=0\, touches the parabola y^{2}=4px\,.Find p and also the point of contact.

The condition that the line lx+my+n=0\, touches y^{2}=4ax\, is

ln=am^{2}\,

From the given line 2x-y+2=0,l=2,m=-1,n=2\,

Therefore,the condition is

p(-1)^{2}=(2)(2)\,

p=4\,

Coordinates of the point of contact are

\left({\frac  {n}{l}},{\frac  {-2pm}{l}}\right)\,

\left({\frac  {2}{2}},{\frac  {-2(4)(-1)}{2}}\right)\,

(1,4)\,

Main Page:Geometry:The Parabola