<p>The equation of a curve is \(y=x^3\). The distinct points \(P\) and \(Q\) lying on the curve have&nbsp;\(x\)-coordinates \(p\) and \(q\) respectively, where&nbsp;\(p\) and \(q\) are real constants.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that the tangent to the curve at \(Q\) passes through point \(P\), show that\[p^2 + pq + cq^2 = 0\] for some constant \(c\) to be determined. Hence, find \(p\) in terms of \(q\).<span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Justify whether there exists a point on the curve such that the normal to the curve at this point cuts the curve again at another point.</span><span class="inline-marks">[4]</span></div></li></ol>

The equation of a curve is \(y=x^3\). The distinct points \(P\) and \(Q\) lying on the curve have \(x\)-coordinates \(p\) and \(q\) respectively, where \(p\) and \(q\) are real constants.

Given that the tangent to the curve at \(Q\) passes through point \(P\), show that\n\[p^2 + pq + cq^2 = 0\]\nfor some constant \(c\) to be determined. Hence, find \(p\) in terms of \(q\).

Given that the tangent to the curve at \(Q\) passes through point \(P\), show that\[p^2 + pq + cq^2 = 0\] for some constant \(c\) to be determined. Hence, find \(p\) in terms of \(q\).

Justify whether there exists a point on the curve such that the normal to the curve at this point cuts the curve again at another point.

NJC IP4 Tangents and Normals Tutorial Q2

NJC IP4 Tangents and Normals Tutorial Q2

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