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A circular running track has the equation x² + y² - 6x - 4y - 12 = 0, with lengths in metres…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A circular running track has the equation $x^2 + y^2 - 6x - 4y - 12 = 0$ , with lengths in metres. A straight cable lies along the line $3x + 4y - 2 = 0$ and cuts across the track. Find the distance of the track centre from the cable, the length of the cable inside the track, and the coordinates of the midpoint of that part of the cable.

A circle with centre (3, 2) and radius 5. A straight red line cuts it in a chord. The perpendicular from the centre to the chord is drawn and labelled d. The two ends of the chord are marked with dots.

Need a reminder? The key ideaShow

A point is inside, on or outside a circle as $S_1 < 0$ , $= 0$ or $> 0$ . A line at distance $d$ from the centre cuts the circle in a chord of length $2\sqrt{r^2 - d^2}$ if $d < r$ . The tangent at $(x_1, y_1)$ is $T = 0$ and the normal passes through the centre.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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