Analytical Geometry

Maths, HSSC-Punjab Second Year. This chapter is split into 42 topics and has 1,082 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Right bisectors of the sides of a triangle 27 questions
2. Special cases of ax2 + 2hxy + by2 = 0 21 questions
3. Interior angles of quadrilaterals 22 questions
4. Family of lines: slope and parallel conditions 26 questions
5. Angle between two lines: theory 22 questions
6. Collinear points and area zero 27 questions
7. Angle between lines joining given points 24 questions
8. Family of lines through the intersection of two lines (theory) 20 questions
9. Family of lines: perpendicular, point and origin conditions 26 questions
10. Joint equation of lines through the origin 28 questions
11. Applications: distance, slope and tilt 27 questions
12. Nature of the lines: h-squared versus ab 29 questions
13. Slope-intercept form (numerical) 29 questions
14. Conditions on constants for pairs of lines 28 questions
15. Parallel and perpendicular lines (conditions) 28 questions
16. Forms of a straight line (definitions) 25 questions
17. Area of a triangle from its vertices 28 questions
18. Area with letters, polygons and shapes 28 questions
19. Applications of pairs of lines through the origin 29 questions
20. Normal form with given p and angle 23 questions
21. Centroid, orthocentre and circumcentre 28 questions
22. Concurrency: finding an unknown constant 27 questions
23. Separating a pair of lines from its joint equation 30 questions
24. Pair of lines perpendicular to a given pair 23 questions
25. Homogeneous expressions and their degree 28 questions
26. Point of concurrency of three lines 28 questions
27. Angle between the lines ax2 + 2hxy + by2 = 0 29 questions
28. Two-point form (numerical) 23 questions
29. Altitudes of a triangle 27 questions
30. Angle from slope m1 to slope m2 22 questions
31. Angle between lines given by their equations 26 questions
32. Interior angles of a triangle 26 questions
33. Right triangles and special quadrilaterals by slopes 27 questions
34. Area of a triangle: derivation and determinant form 21 questions
35. Homogeneous equations and pairs of lines (concepts) 22 questions
36. Applications: roads, plots and stations 29 questions
37. Testing concurrency of three lines 28 questions
38. Medians of a triangle 27 questions
39. Concurrency theorems: proof ideas 24 questions
40. Finding an unknown from a given angle 24 questions
41. Point-slope form (numerical) 21 questions
42. Intercept form (numerical) 25 questions

Sample questions: Analytical Geometry

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Why can the intercept form x/a + y/b = 1 not represent a line passing through the origin?

• A. both intercepts are zero and the fractions x/0 and y/0 are undefined(correct answer)
• B. a line through the origin always has equal positive intercepts
• C. the right-hand side must be zero for such a line
• D. a line through the origin has an infinite slope so no intercepts exist

Answer: A.For a line through the origin the intercepts are a = 0 and b = 0 (a line through the origin that is not an axis), and division by zero is not defined (standard knowledge).

Q2. Which is the correct difference between a median and a right bisector of the same side of a triangle?

• A. a median and a right bisector are the same line in every triangle, whatever the lengths of its sides
• B. a median is perpendicular to the side; a right bisector joins the opposite vertex to the mid-point of the side
• C. a median joins the opposite vertex to the mid-point; a right bisector meets the side at its mid-point at right angles(correct answer)
• D. a median passes through the vertex and is perpendicular to the side; a right bisector is parallel to that side

Answer: C.Both pass through the mid-point of the side, but the median also passes through the opposite vertex while the right bisector is perpendicular to the side; they coincide only in an isosceles triangle.

Q3. After substituting the point of intersection of the first two lines into the third equation, the proof writes the result as a determinant. This works because

• A. the result is exactly the expansion of the determinant along its third row(correct answer)
• B. the product of the three line equations always equals the determinant of coefficients
• C. the value of the determinant equals the area of the triangle formed by the three lines
• D. the determinant is obtained by simply adding the three given line equations together

Answer: A.Multiplying the substituted equation by the common denominator gives a₃(b₁c₂ − b₂c₁) + b₃(a₂c₁ − a₁c₂) + c₃(a₁b₂ − a₂b₁) = 0, the expansion of the 3 × 3 determinant along the third row.

Q4. The right bisector of BC from the proof, x(x₂ − x₃) + y(y₂ − y₃) − (1/2)(y₂² − y₃²) − (1/2)(x₂² − x₃²) = 0, differs from the altitude from A in that

• A. its constant term involves the squares of the coordinates of B and C, not those of A(correct answer)
• B. its x and y coefficients are x₂ + x₃ and y₂ + y₃ instead of the differences
• C. it is parallel to BC instead of being perpendicular to it, since the normal direction is the same
• D. it passes through the vertex A instead of passing through the mid-point of the side BC

Answer: A.Both lines have the same normal direction (x₂ − x₃, y₂ − y₃); the right bisector passes through the midpoint of BC, so its constant term has (x₂² − x₃²)/2 and (y₂² − y₃²)/2.

Q5. Find the value of k if the lines 6x − 2y + 22 = 0, −4x + 4y − 12 = 0 and kx − y + 15 = 0 are concurrent.

• A. 6
• B. −4
• C. 5
• D. 4(correct answer)

Answer: D.Concurrency requires the determinant of the coefficients to be 0; this gives a linear equation in k with solution k = 4 (the third line then passes through (−4, −1)).

Q6. Find the value of k if the lines 3x + 5y − 20 = 0, −5x + 4y − 16 = 0 and −4x + 6y + k = 0 are concurrent.

• A. −23
• B. 24
• C. −22
• D. −24(correct answer)

Answer: D.Concurrency requires the determinant of the coefficients to be 0; this gives a linear equation in k with solution k = −24 (the third line then passes through (0, 4)).

Q7. Find the value of k if the lines −x − y − 5 = 0, 6x + 5y + 26 = 0 and kx + 6y + 21 = 0 are concurrent.

• A. 3
• B. −3(correct answer)
• C. −2
• D. −1

Answer: B.Concurrency requires the determinant of the coefficients to be 0; this gives a linear equation in k with solution k = −3 (the third line then passes through (−1, −4)).

Q8. Find the value of k if the lines 2x − 4y − 2 = 0, −3x + y − 7 = 0 and 4x + 5y + k = 0 are concurrent.

• A. 22(correct answer)
• B. −22
• C. 24
• D. 23

Answer: A.Concurrency requires the determinant of the coefficients to be 0; this gives a linear equation in k with solution k = 22 (the third line then passes through (−3, −2)).

Q9. Find the value of k if the lines 3x + 5y − 13 = 0, −5x − y − 15 = 0 and 3x + 6y + k = 0 are concurrent.

• A. −17
• B. 18
• C. −18(correct answer)
• D. −16

Answer: C.Concurrency requires the determinant of the coefficients to be 0; this gives a linear equation in k with solution k = −18 (the third line then passes through (−4, 5)).

Q10. Find the value of k if the lines 5x + 6y + 22 = 0, 2x − y + 2 = 0 and −5x + 4y + k = 0 are concurrent.

• A. −2(correct answer)
• B. 2
• C. 0
• D. −1

Answer: A.Concurrency requires the determinant of the coefficients to be 0; this gives a linear equation in k with solution k = −2 (the third line then passes through (−2, −2)).