4. For Oxford applicants in Mathematics / Mathematics \& Statistics / Mathematics \& Philosophy, OR those not applying to Oxford, ONLY. Point $A$ is on the parabola $y = \frac { 1 } { 2 } x ^ { 2 }$ at ( $a , \frac { 1 } { 2 } a ^ { 2 }$ ) with $a > 0$. The line $L$ is normal to the parabola at $A$, and point $B$ lies on $L$ such that the distance $| A B |$ is a fixed positive number $d$, with $B$ above and to the left of $A$. [0pt] (i) [6 marks] Find the coordinates of $B$ in terms of $a$ and $d$. [0pt] (ii) [4 marks] Show that in order for $B$ to lie on the parabola, we must have $$a ^ { 2 } d = 2 \left( 1 + a ^ { 2 } \right) ^ { 3 / 2 }$$ (iii) [2 marks] Let $t = a ^ { 2 }$ and express the equality ( $*$ ) in the form $d ^ { 2 / 3 } = f ( t )$ for some function $f$ which you should determine explicitly. [0pt] (iv) [3 marks] Find the minimum value of $f ( t )$. Hence show that the equality ( $*$ ) holds for some real value of $a$ if and only if $d$ is greater than or equal to some value, which you should identify.
marks
4.
For Oxford applicants in Mathematics / Mathematics \& Statistics / Mathematics \& Philosophy, OR those not applying to Oxford, ONLY.
Point $A$ is on the parabola $y = \frac { 1 } { 2 } x ^ { 2 }$ at ( $a , \frac { 1 } { 2 } a ^ { 2 }$ ) with $a > 0$. The line $L$ is normal to the parabola at $A$, and point $B$ lies on $L$ such that the distance $| A B |$ is a fixed positive number $d$, with $B$ above and to the left of $A$.\\[0pt]
(i) [6 marks] Find the coordinates of $B$ in terms of $a$ and $d$.\\[0pt]
(ii) [4 marks] Show that in order for $B$ to lie on the parabola, we must have
$$a ^ { 2 } d = 2 \left( 1 + a ^ { 2 } \right) ^ { 3 / 2 }$$
(iii) [2 marks] Let $t = a ^ { 2 }$ and express the equality ( $*$ ) in the form $d ^ { 2 / 3 } = f ( t )$ for some function $f$ which you should determine explicitly.\\[0pt]
(iv) [3 marks] Find the minimum value of $f ( t )$. Hence show that the equality ( $*$ ) holds for some real value of $a$ if and only if $d$ is greater than or equal to some value, which you should identify.