The length of the chord of the parabola x^2=4y having equation x-√2y+4√2=0 is:

Question

The length of the chord of the parabola $x^2=4y$ having equation $x-\sqrt2y+4\sqrt2=0$ is:

Accepted Answer

Substituting $x=\sqrt2y-4\sqrt2$ into $x^2=4y$: $\sqrt2x^2-4x-16\sqrt2=0$, so $x_1+x_2=2\sqrt2$, $x_1x_2=-16$, $(x_1-x_2)^2=8+64=72$.

Since $x_2-x_1=\sqrt2(y_2-y_1)$:
$$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{(x_2-x_1)^2\left(1+\frac12\right)}=\sqrt{72\cdot\frac32}=\sqrt{108}=6\sqrt3.$$

Solutions · 1