If a line along a chord of the circle 4x^2+4y^2+120x+675=0, passes through the point (-30,…

Question

If a line along a chord of the circle $4x^2+4y^2+120x+675=0$, passes through the point $(-30,0)$ and is tangent to the parabola $y^2=30x$, then the length of this chord is

Accepted Answer

Circle: $x^2+y^2+30x+\frac{675}4=0$, centre $(-15,0)$, $r^2=225-\frac{675}4=\frac{225}4$, $r=\frac{15}2$.

Tangent to $y^2=30x$ ($A=7.5$): $y=mx+\frac{7.5}m$; through $(-30,0)$: $30m=\frac{7.5}m\Rightarrow m^2=\frac14\Rightarrow m=\frac12$.

Line: $x-2y+30=0$. Distance from $(-15,0)$: $\dfrac{|-15+30|}{\sqrt5}=3\sqrt5$.

$$\text{Chord}=2\sqrt{r^2-d^2}=2\sqrt{\frac{225}4-45}=2\sqrt{\frac{45}4}=3\sqrt5.$$

Solutions · 1