The point \(P\) travels along the curve \(C_3\) with its \(x\)-coordinate decreasing at a constant rate of 5 units per second. Find the rate of change of its \(y\)-coordinate when \(x=\dfrac{1}{4}\).[4]
The distance between two points along a curve is the arc-length.\n\nThe arc-length between two points on \(C_3\), where \(p=\alpha\) and \(p=\beta\), is given by the formula \(\left|\int_{\alpha}^{\beta}\sqrt{\left(\dfrac{dx}{dp}\right)^2+\left(\dfrac{dy}{dp}\right)^2}\,dp\right|.\)\n\nFind the total length of the curve from the point \(A(-3,-6)\) to \(B(1,0)\).[3]
Find the equation of the tangent to \(C_3\) at \(p=1\). Show algebraically that this tangent does not cut \(C_3\) again.[5]
Find the cartesian equation of \(C_3\), expressing your answer in the form \(y^2=\mathrm{f}(x)\).[2]
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