Circle intersections with lines/tangents

Geometry and measures

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Does the point $(3, 4)$ lie on the circle $x^{2} + y^{2} = 25$?

2 marks
The circle $(x - 1)^{2} + (y - 2)^{2} = 9$ has centre $(1, 2)$. Does the centre lie on the circle?

2 marks
Find the points where the line $x = 3$ meets $x^{2} + y^{2} = 25$.

3 marks
Find the points where $y = 0$ meets $(x - 2)^{2} + y^{2} = 9$.

3 marks
Find the points where $y = x + 1$ meets $x^{2} + y^{2} = 25$.

4 marks
Find the points where $y = 2x$ meets $x^{2} + y^{2} = 20$.

4 marks
A circle has centre $(0, 0)$ and radius $\sqrt{13}$. Find where the line $y = x - 1$ meets the circle.

5 marks
The line $x + y = 4$ meets the circle $x^{2} + y^{2} = 10$. Find the intersection points and the distance between them. Is the chord a diameter?

6 marks

Mark scheme

Answer. Yes
- $3^{2} + 4^{2} = 25$
- The point lies on the circle.
Answer. No
- At the centre the left side is $0$, not $9$.
- The centre is inside the circle, not on it.
Answer. $(3, 4)$ and $(3, -4)$
- $9 + y^{2} = 25$, so $y^{2} = 16$.
- $y = \pm 4$
Answer. $(5, 0)$ and $(-1, 0)$
- $(x - 2)^{2} = 9$
- $x - 2 = \pm 3$, so $x = 5$ or $x = -1$.
Answer. $(3, 4)$ and $(-4, -3)$
- $x^{2} + (x + 1)^{2} = 25$
- $x^{2} + x - 12 = 0$, so $(x + 4)(x - 3) = 0$.
- The points are $(3, 4)$ and $(-4, -3)$.
Answer. $(2, 4)$ and $(-2, -4)$
- $x^{2} + 4x^{2} = 20$, so $x^{2} = 4$.
- The points are $(2, 4)$ and $(-2, -4)$.
Answer. $(3, 2)$ and $(-2, -3)$
- $x^{2} + (x - 1)^{2} = 13$
- $x^{2} - x - 6 = 0$, so $(x - 3)(x + 2) = 0$.
- The points are $(3, 2)$ and $(-2, -3)$.
Answer. $(1, 3)$ and $(3, 1)$. The distance is $2\sqrt{2}$. It is not a diameter.
- $x^{2} + (4 - x)^{2} = 10$ gives $(x - 1)(x - 3) = 0$.
- The points are $(1, 3)$ and $(3, 1)$.
- Distance $= \sqrt{2^{2} + (-2)^{2}} = 2\sqrt{2}$.
- The midpoint $(2, 2)$ is not the centre, so the chord is not a diameter.