Let [math]F\left(a,0\right)[/math] be the focus and [math]x=-a[/math] be the equation of the directrix of the parabola. Let [math]P\left(x,y\right)[/math] be a point [br]on the parabola. Let [math]B[/math] be the foot of the perpendicular from [math]P[/math] to the directrix. Then the coordinates of the point [math]B=\left(-a,y\right)[/math]. [br]Now since the point on the parabola satisfies [math]PF=PB[/math] we get[math]\sqrt{\left(x-a\right)^2+\left(y-0\right)^2}=\sqrt{\left(x+a\right)^2+\left(y-y\right)^2}[/math][br][math]\Longrightarrow\left(x_{ }-a\right)^2+y^2=\left(x+a\right)^2[/math][br][math]\Longrightarrow x^2-2ax+a^2+y^2=x^2+2ax+a^2[/math][br][math]\Longrightarrow y^2=4ax[/math][br]
Which line is the axis of the parabola?
The parabola is symmetric about which axis?
When [math]a>0[/math] the parabola is concave towards which part of x-axis?
The focus sits on the axis true or false?
When [math]a<0[/math] the focus comes to the positive part of x-axis, true or false?
The focus and the directrix are on opposite sides of the parabola, true/false?
Which point is the vertex of the parabola?
The focus and the directrix are on the same side of the vertex, true or false?
If [math]y^2=x[/math] is the equation of the parabola then check that are correct.
If [math]\left(y-k\right)^2=4a\left(x-h\right)[/math] then what will happen to the vertex ?
The vertex will be shifted to the point [math]\left(h,k\right)[/math]