1.2 Parabola
A parabola is a locus of points \(P\) which are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).
Let the focus be the point \(F(0,P)\) , the vertex be the point \((0,0)\) and the directrix the line \(y = -P\). If \(P(x,y)\) lies on the parabola then \(FP = PD\).
\[\implies \hspace {0.4cm} \sqrt {(x- 0)^2 + (y - P)^2} = \sqrt {(x - x)^2 + (y-(-p))^2}\]
\[\implies \hspace {0.5cm} \Big (\sqrt {x^2 + (y - P)^2}\Big )^2 = (y+P)^2\]
\[\implies \hspace {0.5cm} x^2 + y^2 - 2Py + P^2 = y^2 + 2Py + P^2\]
\[\boxed {x^2 = 4Py}\]
\(\implies \hspace {0.3cm} x^2 = 4Py\hspace {0.3cm}\) which is the standard equation of a parabola whose focus \(F(0,P)\) and directrix \(y = -P\).
The standard equation of a parabola which opens downward with focus \(F(0, -P)\) and directrix \(y = P\) is given by \(\boxed {x^2 = -4Py}\)
Derive the equation \(x^2 = -4Py\).
\(\star \) The standard equation of a parabola which opens to the right along the \(x-\)axis with focus \(F(0,P)\) and directrix on the line \(x = -P\) is given by \(\boxed {y^2 = 4Px}\)
The standard equation of the parabola which open to the left with focus \(F(-P,0)\) and directrix on the line \(x = P\) is given by \(\boxed { y^2 = - 4Px}\)
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