A curve is defined by the parametric equations $x=a{{t}^{2}},\,y=2at.$ Show that the equation of the

Question

A curve is defined by the parametric equations $x=a{{t}^{2}},\,y=2at.$ Show that the equation of the normal to the curve at the point $(a{{t}^{2}},\,2at)$ is at $y+tx=a{{t}^{3}}+2at.$ If the normal to the curve at the point $(a{{t}^{2}},\,2at)$ meets the curve again at the point $(a{{q}^{2}},\,2aq),$ show that ${{t}^{2}}+qt+2=0.$ Deduce that ${{q}^{2}}$ cannot be less than $8$.

Accepted Answer

Normal: y + tx = at³ + 2at; t² + qt + 2 = 0; q² ≥ 8

2007 AJC P1 Q12

2007 AJC P1 Q12

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