T4T2 Review — Coordinate Geometry

Mixed review covering L43 (Distance, Midpoint & Gradient), L44 (Equations of Lines) and L45 (Circles and Coordinate Geometry).

1. Distance and midpoint. Fluency

   • (a) Find the distance between A(1, 4) and B(7, 12).
   • (b) Find the midpoint M of A(1, 4) and B(7, 12).
   • (c) P is the midpoint of C(3, −1) and D. The midpoint is (5, 2). Find D.
   • (d) Show that A(2, 1), B(5, 5) and C(8, 9) are collinear (lie on the same line).

2. Gradient. Fluency

   • (a) Find the gradient of the line through (2, −3) and (6, 5).
   • (b) A line has gradient 3/4. State the gradient of a line perpendicular to it.
   • (c) Are the lines through (0, 1) and (4, 3), and through (1, 0) and (3, 4), perpendicular?
   • (d) The line through (a, 3) and (4, −1) has gradient −2. Find a.

3. Equations of lines. Fluency

   • (a) Write the equation of the line with gradient 3 and y-intercept −4.
   • (b) Write the equation of the line through (0, 5) and (3, 11).
   • (c) Write the equation of the line through (4, 1) with gradient −1/2.
   • (d) Find the x-intercept and y-intercept of 3x − 2y = 12.

4. Circle equations. Fluency

   • (a) Write the equation of a circle with centre (−2, 5) and radius 4.
   • (b) State the centre and radius of (x − 3)² + (y + 1)² = 81.
   • (c) Is the point (5, −1) on, inside or outside (x − 3)² + (y + 1)² = 9?
   • (d) Find the centre and radius of x² + y² − 6x + 10y + 18 = 0.

5. Parallel and perpendicular lines. Understanding

   • (a) Find the equation of the line through (2, 5) parallel to y = 4x − 1.
   • (b) Find the equation of the line through (6, 0) perpendicular to 2x + y = 5.
   • (c) The perpendicular bisector of the segment from A(1, 3) to B(5, 7). Find its equation.
   • (d) The lines 2x − y = 3 and ax + 3y = 1 are parallel. Find a.

6. Intersection of lines. Understanding

   • (a) Find the intersection of y = 2x − 1 and y = −x + 8.
   • (b) Find the intersection of 3x + 2y = 12 and x − y = 1.
   • (c) Show that the three lines y = x + 1, y = 2x − 1 and 3x − y = 5 are concurrent (all meet at one point).
   • (d) Find the area of the triangle formed by y = 0, x = 0 and 2x + 3y = 12.

7. Tangents to circles. Understanding

   • (a) Find the equation of the tangent to x² + y² = 25 at P(3, −4).
   • (b) Find the equation of the tangent to (x−1)² + (y−2)² = 5 at P(3, 3).
   • (c) The tangent to x² + y² = r² at (a,b) is ax + by = r². Use this to find the tangent to x² + y² = 50 at (5, −5).
   • (d) A tangent from the external point (10, 0) touches the circle x² + y² = 50. Find the length of the tangent (from the external point to the tangent point).

8. Circles and lines — intersections. Understanding

   • (a) Find where the line y = x − 1 meets the circle x² + y² = 25.
   • (b) Show that the line y = x + 6 is tangent to the circle x² + y² = 18.
   • (c) How many intersections does y = 5 have with x² + y² = 4? Justify without solving.
   • (d) Find the chord length where x + y = 4 intersects x² + y² = 10.

9. Coordinate geometry proof. Problem Solving

   The quadrilateral ABCD has vertices A(0, 0), B(4, 0), C(6, 4), D(2, 4).

   • (a) Calculate all four side lengths. What type of quadrilateral is ABCD?
   • (b) Show that the diagonals of ABCD bisect each other (find their midpoints).
   • (c) Are the diagonals perpendicular? Use gradients to decide.
   • (d) Find the equation of the line through A and C. What is the y-value on this line when x=3?

10. Circle through three points. Problem Solving

    Three points lie on a circle: P(1, 0), Q(3, 0), R(2, 3).

    • (a) Find the perpendicular bisector of PQ.
    • (b) Find the perpendicular bisector of QR.
    • (c) Find the centre of the circle by solving the two perpendicular bisectors simultaneously.
    • (d) Write the equation of the circle and verify that P, Q and R lie on it.

11. Real-world coordinate problem. Problem Solving

    A mobile phone tower is modelled as the centre of a circle with coverage radius of 4 km. The tower is at T(3, 5) on a map where 1 unit = 1 km. A road runs along y = 2x − 1.

    • (a) Write the equation of the coverage circle.
    • (b) Find where the road intersects the coverage boundary (solve simultaneously).
    • (c) Find the length of road that is within the coverage area.
    • (d) The nearest point on the road to the tower: find it (perpendicular from T to the road), and state whether the tower is inside or outside the coverage circle of a second tower at S(7, 5) with radius 3 km.

12. Extended: coordinate geometry theorem. Problem Solving

    Prove using coordinate geometry that the angle in a semicircle is always 90°. Use the circle x² + y² = r² with diameter endpoints A(−r, 0) and B(r, 0), and an arbitrary point P(x⊂0;, y⊂0;) on the circle.

    • (a) Write expressions for the gradient of PA and the gradient of PB.
    • (b) Find the product m⊂PA; × m⊂PB;.
    • (c) Use the fact that P lies on the circle (x⊂0;² + y⊂0;² = r²) to simplify.
    • (d) Conclude the proof. What condition ensures this proof is valid?