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MathsMediumClass 12

Locus Problems

Straight Lines

5

JEE Qs

8%

Hard

75

min

Always follow the systematic approach: assume (h, k), apply all conditions algebraically, eliminate other variables, and then replace (h, k) with (x, y) to find the locus equation.

🧮 Key Formulas

Distance between (x1, y1) and (x2, y2): d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Section formula (internal division): ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n))
Midpoint formula: ((x1 + x2)/2, (y1 + y2)/2)
Slope of line through (x1, y1) and (x2, y2): m = (y2 - y1)/(x2 - x1)
Equation of a line (point-slope form): y - y1 = m(x - x1)
Distance of a point (x0, y0) from a line Ax + By + C = 0: |Ax0 + By0 + C| / sqrt(A^2 + B^2)

✅ Key Points for JEE

  • 1Assume the coordinates of the variable point whose locus is to be found as (h, k).
  • 2Translate the given geometric condition(s) into algebraic equations involving h, k, and other fixed/variable points using relevant coordinate geometry formulas.
  • 3If the condition involves other variable points, eliminate their coordinates to obtain an equation solely in terms of h, k, and constants.
  • 4Replace h with x and k with y in the final equation to get the equation of the locus.
  • 5Recognize standard curves (straight line, circle, parabola, ellipse, hyperbola) from the derived equation; this helps verify the answer and understand the geometry.

⚠️ Common Mistakes

  • ✕Making algebraic errors during simplification, squaring, or expansion, which leads to an incorrect final equation.
  • ✕Failing to eliminate all auxiliary variable coordinates from the equation, resulting in a locus equation that is not solely in terms of (h, k) and constants.
  • ✕Incorrectly translating the geometric description (e.g., perpendicular, equidistant) into the corresponding algebraic condition.
  • ✕Prematurely assuming the shape of the locus (e.g., assuming it's a circle) instead of systematically deriving the equation.

📝 Practice Questions

See all

Q20.Two equal sides of an isosceles triangle are along −x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is : (1) −2√10 (2) 12 (3) 6 (4) −6

2025·MCQHard

Q15.Let ABC be a triangle formed by the lines 7x −6y + 3 = 0, x + 2y −31 = 0 and 9x −2y −19 = 0. Let the point (h, k) be the image of the centroid of ΔABC in the line 3x + 6y −53 = 0. Then h2 + k2 + hk is equal to: (1) 47 (2) 37 (3) 36 (4) 40 is:

2025·MCQMedium

Q9. If α and β are the roots of the equation 2z2 −3z −2i = 0, where i = √−1, then α19+β19+α11+β11 α19+β19+α11+β11 16 ⋅Re ⋅lm is equal to ( α15+β15 ) ( α15+β15 ) 2025 (24 Jan Shift 1) JEE Main Previous Year Paper (1) 441 (2) 398 (3) 312 (4) 409

2025·MCQHard

Q6. Let the points ( 112 , α) lie on or inside the triangle with sides x + y = 11, x + 2y = 16 and 2x + 3y = 29. Then the product of the smallest and the largest values of α is equal to : (1) 44 (2) 22 (3) 33 (4) 55

2025·MCQMedium

Q65.Let A(−1, 1) and B(2, 3) be two points and P be a variable point above the line AB such that the area of △PAB is 10 . If the locus of P is ax + by = 15, then 5a + 2 b is : (1) 6 (2) −65 (3) 4 (4) −125

2024·MCQMedium

Q64.Let a variable line of slope m > 0 passing through the point (4, −9) intersect the coordinate axes at the points A and B. The minimum value of the sum of the distances of A and B from the origin is JEE Main 2024 (06 Apr Shift 1) JEE Main Previous Year Paper (1) 30 (2) 25 (3) 15 (4) 10

2024·MCQHard

NCERT Chapters

  • Class 10 Maths Ch 7: Coordinate Geometry
  • Class 11 Maths Ch 10: Straight Lines
  • Class 11 Maths Ch 11: Conic Sections