91Ó°ÊÓ

The questions are all straightforward. Find the angle between the curves \(x^{2}+y^{2}=4\) and \(5 x^{2}+y^{2}=5\) at their point of intersection for which \(x\) and \(y\) are positive.

Find the equation of the normal to the curve \(y=\frac{2 x}{x^{2}+1}\) at the point \((3,0 \cdot 6)\) and the equation of the tangent at the origin.

Find the equations of the tangent and normal to the curve \(4 x^{3}+4 x y+y^{2}=4\) at \((0,2)\), and find the coordinates of a further point of intersection of the tangent and the curve.

The questions are all straightforward. Find the equations of the tangent and normal to the curve \(y^{2}=11-\frac{10}{4-x}\) at the point \((6,4)\)

The questions are all straightforward. The parametric equations of a function are \(x=2 \cos ^{3} \theta, y=2 \sin ^{3} \theta\). Find the equation of the normal at the point for which \(\theta=\frac{\pi}{4}=45^{\circ}\).

Obtain the equations of the tangent and normal to the ellipse \(\frac{x^{2}}{169}+\frac{y^{2}}{25}=1\) at the point \((13 \cos \theta, 5 \sin \theta)\). If the tangent and normal meet the \(x\)-axis at the points \(\mathrm{T}\) and \(\mathrm{N}\) respectively, show that ON.OT is constant, \(\mathrm{O}\) being the origin of coordinates.

The questions are all straightforward. If \(x=1+\sin 2 \theta, y=1+\cos \theta+\cos 2 \theta\), find the equation of the tangent at \(\theta=60^{\circ} .\)

If \(x^{2} y+x y^{2}-x^{3}-y^{3}+16=0\), find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) in its simplest form. Hence find the equation of the normal to the curve at the point \((1,3)\).

The questions are all straightforward. Find the radius of curvature and the coordinates of the centre of curvature at the point \(x=4\) on the curve whose equation is \(y=x^{2}+5 \ln x-24\).

Find the radius of curvature of the catenary \(y=c \cosh \left(\frac{x}{c}\right)\) at the point \(\left(x_{1}, y_{1}\right)\)