If the curve x^2+2y^2=2 intersects the line x+y=1 at two points P and Q, then the angle su…

Question

If the curve $x^2+2y^2=2$ intersects the line $x+y=1$ at two points $P$ and $Q$, then the angle subtended by the line segment $PQ$ at the origin is:

Accepted Answer

Homogenize $x^2+2y^2=2$ using $x+y=1$: $x^2+2y^2-2(x+y)^2=0 \implies -x^2-4xy=0 \implies -x(x+4y)=0$

This pair of lines represents $OP$ and $OQ$: $x=0$ and $x+4y=0$ (i.e. $y=-\dfrac x4$).

The angle between $x=0$ (the $y$-axis) and $y=-\dfrac x4$ (slope $-\dfrac14$, inclination $\pi-\tan^{-1}\dfrac14$ from positive $x$-axis) is
$$\frac{\pi}2+\tan^{-1}\frac14$$

Solutions · 1