HardMathematicsHigh School (11–12)Coordinate Geometry › Circles and Conic SectionsNumerical

A circle x² + y² = 25 and the line x + y = 1 meet at two points…

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

A circle $x^2 + y^2 = 25$ and the line $x + y = 1$ meet at two points. A new circular path is laid out through both meeting points and through the origin, with lengths in metres. Find the coordinates of the centre of this path, written as two equal coordinates, and its radius.

Need a reminder? The key ideaShow

$S + \lambda L = 0$ and $S_1 + \lambda S_2 = 0$ give the circles through the meeting points. A circle touching $L = 0$ at $(x_1, y_1)$ is $(x - x_1)^2 + (y - y_1)^2 + \lambda L = 0$ . For a locus, write the rule for $(h, k)$ and replace $(h, k)$ by $(x, y)$ .

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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