Discussion questions · Section 2 of 3

The equation of a closed curve is $(x+2y)^2+3(x-y)^2=27$.

(i) Show by differentiation that its gradient is $\dfrac{dy}{dx}=\dfrac{y-4x}{7y-x}$.

(ii) Find the equations of the tangents parallel to (a) the x-axis, (b) the y-axis.

The curve $C$ has equation $2^{-y}=x$. Its point $A$ has x-coordinate $a>0$. Show that $dy/dx=-1/(a\ln2)$ at $A$, and find the equation of the tangent there.

Hence find the tangent to $C$ that passes through the origin. The line $y=mx$ intersects $C$ at two distinct points. Write down the range of $m$.

The curve has equation $y^2-xy=-1$.

(i) Find all tangent equations parallel to the y-axis.

(ii) State and justify whether the curve has any stationary points.

(iii) Find the area bounded by the axes and the normal at the point where $y=2$.

The right-angled triangle $ABC$ has $AB=c$ cm, $BC=a$ cm and $AC=b$ cm, with right angle at $C$ and fixed area $100\text{ cm}^2$.

Express $b$ in terms of $a$. Given that $BC$ increases at $3\text{ cm s}^{-1}$, find the rate of change of $c$ when $a=20$.

The diagram shows rectangle $PQRS$ inscribed in a semicircle of radius $10$ cm, with $P,Q$ on its diameter and $S,R$ on its arc. Given $PQ=x$ cm, find the perimeter $p$ cm in terms of $x$.

Show that its area is $A=\dfrac{x}{2}\sqrt{400-x^2}\text{ cm}^2$. Find the exact value of $p$ when $A$ has its stationary value.

A cone of radius $r$ and height $h$ has volume $\tfrac13\pi r^2h$. A sphere of fixed radius $a$ and centre $O$ is inscribed in a right cone, touching its base and curved surface, as shown.

(i) Show that $r^2=\dfrac{a^2h}{h-2a}$.2 marks

(ii) Use differentiation to find the minimum volume of the cone in terms of $a$, proving it is a minimum.6 marks