Find the coordinates of the point(s) to the curve $2{{x}^{2}}+xy-{{y}^{2}}=9$ at which the tangent

Question

Find the coordinates of the point(s) to the curve $2{{x}^{2}}+xy-{{y}^{2}}=9$ at which the tangent is parallel to the $y$ – axis.[5] The curve $C$ has parametric equations $x={{t}^{2}}+t,\mathrm{ }y=4-t.$ The point $P$ on the curve has parameter $p$. Show that the equation of the tangent at $P$ is $(2p+1)(4-p-y)=x-{{p}^{2}}-p$.[2] Hence, show that every tangent to the curve $C$ does not meet the curve again.[3]

Accepted Answer

(a) $(2,1)$ and $(-2,-1)$ (b)(i) $(2p+1)(4-p-y)=x-p^2-p$ (b)(ii) $(t-p)^2=0$, so there is no second distinct intersection

2013 SRJC P1 Q10

2013 SRJC P1 Q10

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