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Math · 27 · Geometry and Trigonometry

Advanced Coordinate Geometry

Notes

Key points

  • Distance = Pythagoras on the horizontal and vertical gaps.
  • Midpoint = average the x's, average the y's.
  • Line meets parabola or circle: substitute the line into the other equation.
  • The resulting quadratic's discriminant counts intersections: 2, 1 (tangent) or 0.
  • “Exactly one intersection”: set b2−4ac=0b^2-4ac=0 and solve for the constant.
  • Solve for x, then plug back into the line to get y.

Formulas

Distanced=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}Not on the reference sheet
MidpointM=(x1+x22, y1+y22)M=\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)
Missing endpointx2=2Mx−x1,y2=2My−y1x_2=2M_x-x_1,\quad y_2=2M_y-y_1
Line meets parabolamx+k=ax2+bx+c ⇒ ax2+(b−m)x+(c−k)=0mx+k=ax^2+bx+c\ \Rightarrow\ ax^2+(b-m)x+(c-k)=0
Number of intersectionsD=b2−4acD=b^2-4acPositive: two; zero: one (tangent); negative: none

Common traps

  • Missing endpoint: averaging instead of x2=2Mx−x1x_2=2M_x-x_1.
  • Sign slips inside the square: (1−(−3))2=16(1-(-3))^2=16, not 4.
  • Stopping at x when the question asks for the intersection's y-coordinate.

SAT tip

Graph both in Desmos and click the intersections. With an unknown constant and “one solution”, use the discriminant.

Test yourself

10 official SAT questions on Advanced Coordinate Geometry, easiest first.

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