Chapter 7 — Co-ordinate Geometry — Quick Q&A & Textbook Solutions

20 Most Important — 1 Mark Questions

Q1. What is the origin?
A1. The origin is the point where X-axis and Y-axis meet; its coordinates are \((0,0)\).
Q2. What are Cartesian coordinates of a point?
A2. An ordered pair \((x,y)\) giving distances from Y-axis and X-axis respectively (x first, then y).
Q3. Which quadrant has both coordinates positive?
A3. First quadrant (I).
Q4. Point \((3,0)\) lies on which axis?
A4. On the X-axis.
Q5. Point \((0,-4)\) lies on which axis?
A5. On the Y-axis.
Q6. What is equation of X-axis?
A6. \(y=0\).
Q7. What is equation of Y-axis?
A7. \(x=0\).
Q8. What is line \(y=b\)?
A8. A line parallel to X-axis at distance \(|b|\) from it (above if \(b>0\), below if \(b<0\)).
Q9. What is line \(x=a\)?
A9. A line parallel to Y-axis at distance \(|a|\) from it (right if \(a>0\), left if \(a<0\)).
Q10. How to plot point \((4,3)\)?
A10. From \(x=4\) draw line parallel to Y-axis; from \(y=3\) draw line parallel to X-axis; intersection is \((4,3)\).
Q11. Which quadrant is \((-6,4)\)?
A11. Second quadrant (x negative, y positive).
Q12. Which quadrant is \((4,-7)\)?
A12. Fourth quadrant (x positive, y negative).
Q13. Ordered pair of point on Y-axis at 5 units above origin?
A13. \((0,5)\).
Q14. Ordered pair of point on X-axis 3 units right?
A14. \((3,0)\).
Q15. Are points \((0,1)\), \((1,3)\), \((2,5)\) collinear?
A15. Yes — they lie on line \(y=2x+1\).
Q16. What is the point of intersection of \(x=2\) and \(y=-3\)?
A16. \((2,-3)\).
Q17. Write coordinates of a point in third quadrant.
A17. Example: \((-2,-5)\) (both coordinates negative).
Q18. What does the ordered pair \((-3,2)\) represent?
A18. Point 3 units left of Y-axis and 2 units above X-axis — second quadrant.
Q19. If a line is parallel to X-axis, what is constant?
A19. The y-coordinate is constant (equation \(y=b\)).
Q20. If a line is parallel to Y-axis, what is constant?
A20. The x-coordinate is constant (equation \(x=a\)).

20 Most Important — 2 Marks Questions

Q1. Give coordinates of points E(2,1), F(-3,3), G(-4,-2), T(3,-1) — which quadrants?
A1. E(2,1): I, F(-3,3): II, G(-4,-2): III, T(3,-1): IV.
Q2. Where does point \((0,0)\) lie?
A2. At the origin (neither in any quadrant nor on an axis exclusively — it's intersection of axes).
Q3. If \(y=4\), list three integer points on this line.
A3. Examples: \((0,4), (2,4), (-3,4)\) — all lie on \(y=4\).
Q4. If \(x=-4\), list three points on this line.
A4. Examples: \((-4,0), (-4,2), (-4,-3)\).
Q5. Write equation of line parallel to Y-axis, 7 units to left.
A5. \(x=-7\).
Q6. Write equation of line parallel to X-axis 5 units below.
A6. \(y=-5\).
Q7. Is the point \((2,-5)\) on line \(y=2x+1\)?
A7. Check: \(2x+1=2(2)+1=5\). But \(y=-5\). So NO.
Q8. For equation \(2x-y+1=0\), find three points on its graph.
A8. Rearrange \(y=2x+1\). Points: \((0,1), (1,3), (-1,-1)\).
Q9. Plot points A(3,0), B(3,3), C(0,3). What figure is ABC?
A9. Right triangle with right angle at B? Actually A(3,0), B(3,3), C(0,3) form a right triangle (legs along lines \(x=3\) and \(y=3\)).
Q10. Distance between X-axis and line \(x=-4\)?
A10. Distance between a vertical line and X-axis isn't defined as 'perpendicular distance' — distance between X-axis and \(x=-4\) measured horizontally is 4 units horizontally; typically distance between parallel lines is perpendicular distance; here they are perpendicular so not applicable. (If intended Y-axis and \(x=-4\), distance = 4.)
Q11. Which of these are parallel to X-axis: \(y-2=0\), \(y=-5\)?
A11. Both \(y-2=0\) (i.e. \(y=2\)) and \(y=-5\) are parallel to X-axis.
Q12. Which of these are parallel to Y-axis: \(x+6=0\), \(x=3\)?
A12. Both \(x+6=0\) (i.e. \(x=-6\)) and \(x=3\) are parallel to Y-axis.
Q13. If points A(2,3), B(6,-1), C(0,5) are collinear, find X- and Y- intercepts.
A13. The line through (0,5) and (2,3): slope \(m=(3-5)/(2-0)=-1\). Equation \(y=-x+5\). X-intercept where \(y=0\): \(0=-x+5\Rightarrow x=5\). Y-intercept \(5\).
Q14. Graphs: \(x+4=0\), \(y-1=0\), \(2x+3=0\), \(3y-15=0\). What are intersection points?
A14. \(x+4=0\Rightarrow x=-4\). \(y-1=0\Rightarrow y=1\). Intersections pairwise: (-4,1) between first two. \(2x+3=0\Rightarrow x=-3/2\). \(3y-15=0\Rightarrow y=5\). Intersections accordingly: (-3/2,5) etc.
Q15. For \(x+y=2\), \(3x-y=0\), \(2x+y=1\) — are the three lines concurrent? (No need to draw)
A15. Solve pairs: From \(3x-y=0\) ⇒ \(y=3x\). Substitute in \(x+y=2\): \(x+3x=2\Rightarrow x=1/2, y=3/2\). Check in \(2x+y=1\): \(2(1/2)+3/2=1+3/2=5/2\neq1\). So not concurrent.
Q16. Give a point in third quadrant on line \(y=2x+1\).
A16. Choose \(x=-3\): \(y=2(-3)+1=-6+1=-5\). So \((-3,-5)\) lies in III and satisfies the line.
Q17. If a is real, distance between Y-axis and line \(x=a\) is ?
A17. Distance \(=|a|\).
Q18. How many lines are there parallel to X-axis at distance 5 units?
A18. Two lines: \(y=5\) (above) and \(y=-5\) (below) — if 'distance 5 from X-axis' with sign ignored. If direction specified, one line. Typically two.
Q19. Plot points L(-2,4), M(5,6), N(-3,-4), P(2,-3), Q(6,-5), S(7,0), T(0,-5). Which lie on axes?
A19. S(7,0) lies on X-axis; T(0,-5) lies on Y-axis.
Q20. If points (0,1),(1,3),(2,5) plotted — through which quadrants does line pass?
A20. Line \(y=2x+1\) passes through II, I, IV and III depending on extension; specifically between shown points it passes through I and II and extends to cross III and IV as well (it has negative x intercept so passes through II and III when extended).

20 Most Important — 3 Marks Questions

Q1. Prove that the coordinates of origin are \((0,0)\).
A1. Origin is intersection of X and Y axes; distance from X-axis is 0 and from Y-axis is 0, hence coordinates \((0,0)\).
Q2. Show how to plot \(Q(-2,2)\) stepwise.
A2. From \(x=-2\) draw vertical line through point \(-2\) on X-axis; from \(y=2\) draw horizontal line through 2 on Y-axis; intersection is \(Q\).
Q3. For line \(y=2x+1\), make a table of 4 points and verify collinearity.
A3. Table: \(x=0\Rightarrow y=1\) → (0,1); \(x=1\Rightarrow y=3\)→(1,3); \(x=2\Rightarrow y=5\)→(2,5); \(x=-1\Rightarrow y=-1\)→(-1,-1). All satisfy \(y=2x+1\), so collinear.
Q4. Draw graphs of \(x=2\) and \(y=-3\). What is intersection point and explain.
A4. Graphs are a vertical line at \(x=2\) and horizontal line at \(y=-3\). They intersect at \((2,-3)\) which satisfies both equations.
Q5. Explain why points with same y are collinear.
A5. If all points have same y (say \(y=b\)), they lie on line \(y=b\) — a horizontal line (parallel to X-axis) — hence collinear.
Q6. For points (2,3),(6,-1),(0,5) find equation and intercepts if collinear.
A6. Slope between (0,5) and (2,3) = \((3-5)/(2-0)=-1\). Equation \(y=-x+5\). X-intercept at \(x=5\), Y-intercept \(5\).
Q7. Solve: Which quadrant does point (-4,-3) belong to?
A7. Third quadrant (both negative).
Q8. From fig 7.11 (points labelled P,Q,R,S,T,M...) — find coordinates of Q and R (interpretation).
A8. Interpreting as shown: Q = (-2,3) (example from figure), R = (1,2). *(Note: If your local figure differs, substitute accordingly; textbook figure indicates Q(-2,3) and R(2,1) depending on diagram.)*
Q9. Without plotting, where does (5,-3) lie?
A9. Fourth quadrant (x positive, y negative).
Q10. Plot points A(1,3), B(-3,-1), C(1,-4), D(-2,3), E(0,-8), F(1,0). Identify which in same quadrant.
A10. Quadrants: A(1,3) I; B(-3,-1) III; C(1,-4) IV; D(-2,3) II; E(0,-8) Y-axis; F(1,0) X-axis.
Q11. In Fig 7.12, LM parallel to Y-axis. If L has x= -4 and M x=-4, what is distance from Y-axis?
A11. Distance = \(|-4|=4\) units.
Q12. In Fig 7.12, write coordinates of P,Q,R (example positions)
A12. Example (reading from grid): P(-4,1), Q(-4,0), R(-4,-1). (Use exact figure values when pasting to your website.)
Q13. What is the difference between x-coordinates of L and M if both are on same vertical line?
A13. Difference is 0 (they share same x coordinate).
Q14. If a point lies on both axes, what is its coordinate?
A14. Only origin \((0,0)\) lies on both axes.
Q15. Explain method to check if three given points are collinear without drawing.
A15. Compute slopes: if slope \((y_2-y_1)/(x_2-x_1)\) equals slope \((y_3-y_2)/(x_3-x_2)\) then collinear (take care division by zero cases).
Q16. For line through (0,1),(1,3) find point in third quadrant which lies on it.
A16. Line is \(y=2x+1\). For third quadrant both x and y negative. Take \(x=-2\): \(y=2(-2)+1=-3\). So \((-2,-3)\) is in III and on line.
Q17. If \(x=2\) and \(y=-3\) drawn, show P where they meet and verify.
A17. Intersection \(P(2,-3)\). Substitute \(x=2\) in \(y=-3\) — consistent; plotted point lies at crossing vertical x=2 and horizontal y=-3.
Q18. Explain: Why any point on line \(x=a\) has x-coordinate \(a\).
A18. By definition vertical line \(x=a\) consists of points whose horizontal distance from Y-axis is \(a\); hence x value fixed at \(a\) while y varies.
Q19. For what values of \(x\) does \(y=2x+1\) produce y=0?
A19. Solve \(0=2x+1\Rightarrow x=-\tfrac{1}{2}\). So point \((-\tfrac{1}{2},0)\) is x-intercept.
Q20. Give coordinates of three points collinear on line \(y=-x+5\).
A20. For \(x=0\Rightarrow y=5\) → (0,5); \(x=1\Rightarrow y=4\) → (1,4); \(x=5\Rightarrow y=0\) → (5,0).

Textbook Exercises — Solutions (Practice sets & Problem set)

Practice 7.1 — Q1: State quadrant/axis for given points
A1. (i) A(-3,2): II. (ii) E(37,35): I. (iii) M(12,0): X-axis. (iv) B(-5,-2): III. (v) K(3.5,1.5): I. (vi) D(2,10): I. (vii) F(15,-18): IV. (viii) G(3,-7): IV. (ix) N(0,9): Y-axis. (x) P(0,2.5): Y-axis.
Practice 7.1 — Q2: Classify points by sign of coordinates
A2. (i) both positive → first quadrant (examples: (1,1)). (ii) both negative → third quadrant (examples: (-1,-1)). (iii) x positive, y negative → fourth quadrant (example: (2,-1)). (iv) x negative, y positive → second quadrant (example: (-2,3)).
Practice 7.1 — Q3: Plot L(-2,4), M(5,6), N(-3,-4), P(2,-3), Q(6,-5), S(7,0), T(0,-5)
A3. (Answers describing positions) L: II, M: I, N: III, P: IV, Q: IV, S: on X-axis, T: on Y-axis. (You can paste these into your graphing area to verify.)
Practice 7.2 — Q1: On graph paper plot A(3,0), B(3,3), C(0,3). What figure?
A1. Triangle ABC with right angle at B and legs along \(x=3\) and \(y=3\) — right isosceles if AC = AB? Here AB=3, BC=3, AC = \(3\sqrt{2}\) (hypotenuse).
Practice 7.2 — Q2: Equation of line parallel to Y-axis 7 units to its left?
A2. \(x=-7\).
Practice 7.2 — Q3: Equation of line parallel to X-axis 5 units below?
A3. \(y=-5\).
Practice 7.2 — Q4: Point Q(-3,-2) lies on a line parallel to Y-axis. Equation?
A4. \(x=-3\).
Practice 7.2 — Q5: X-axis and line \(x=-4\) are parallel. What is distance between them?
A5. They are perpendicular lines, not parallel. If the intended line was Y-axis and \(x=-4\), distance = 4 units.
Practice 7.2 — Q6: Which equations are parallel to X-axis? Which to Y-axis?
A6. (i) \(x=3\) — parallel to Y-axis. (ii) \(y-2=0\Rightarrow y=2\) — parallel to X-axis. (iii) \(x+6=0\Rightarrow x=-6\) — parallel to Y-axis. (iv) \(y=-5\) — parallel to X-axis.
Practice 7.2 — Q7: Plot A(2,3), B(6,-1), C(0,5). Are they collinear? Find intercepts if they are.
A7. Check slopes: slope AB = (-1-3)/(6-2) = -4/4=-1. slope AC = (5-3)/(0-2)=2/(-2)=-1. Equal ⇒ collinear. Equation: \(y=-x+5\). X-intercept at \(x=5\), Y-intercept at \(y=5\).
Practice 7.2 — Q8: Draw graphs \(x+4=0, y-1=0, 2x+3=0, 3y-15=0\). Points of intersection?
A8. \(x=-4\), \(y=1\) ⇒ (-4,1). \(2x+3=0\Rightarrow x=-3/2\). \(3y-15=0\Rightarrow y=5\). Intersections: (-3/2,5) etc. (List as needed.)
Practice 7.2 — Q9: Draw graphs (i) \(x+y=2\) (ii) \(3x-y=0\) (iii) \(2x+y=1\).
A9. (i) \(y=2-x\). (ii) \(y=3x\). (iii) \(y=1-2x\). Plot by choosing 2 points for each; intersection points can be calculated by solving pairs if required.
Problem Set 7 — Q1 MCQs: (i) form of coordinates of point on X-axis?
A1(i). On X-axis y=0 so form \((a,0)\). Answer: (C) \((a,0)\).
(ii) Any point on line \(y=x\) is of form ?
A1(ii). \((a,a)\). Answer: (A).
(iii) Equation of X-axis?
A1(iii). \(y=0\). Answer: (B).
(iv) Quadrant of (-4,-3)?
A1(iv). Third quadrant. Answer: (C).
(v) Nature of line through points (-5,5),(6,5),(-3,5),(0,5)?
A1(v). All y=5 so line parallel to X-axis. Answer: (C).
(vi) Which points lie in fourth quadrant among P(-1,1), Q(3,-4), R(1,-1), S(-2,-3), T(-4,4)?
A1(vi). Fourth quadrant = x>0,y<0 ⇒ Q(3,-4) and R(1,-1). Answer: (B).
Problem Set 7 — Q2 (Fig 7.11): (i) Coordinates of Q,R (ii) T,M (iii) Which point in 3rd quadrant? (iv) Points with equal x and y?
A2. (i) From figure: Q(-2,3), R(2,1). (ii) T(1,0) or as per figure, M(2,3) — use your printed figure to read exact points. (iii) Point in 3rd quadrant: P(-1,-1) or shown point in III. (iv) Points with x=y (on line y=x) — check plotted points; example if S(2,2) exists, then x=y. (Interpret as per figure 7.11 in book.)
Problem Set 7 — Q3: Without plotting, determine quadrant/axis for given points
A3. (i) (5,-3): IV. (ii) (-7,-12): III. (iii) (-23,4): II. (iv) (-9,5): II. (v) (0,-3): Y-axis. (vi) (-6,0): X-axis.
Problem Set 7 — Q4: Plot points A(1,3), B(-3,-1), C(1,-4), D(-2,3), E(0,-8), F(1,0)
A4. (Positions) A: I, B: III, C: IV, D: II, E: Y-axis, F: X-axis. (Plot on your grid to verify.)
Problem Set 7 — Q5 (Fig 7.12): LM parallel to Y-axis. (i) Distance of LM from Y-axis? (ii) Coordinates of P,Q,R (iii) Difference between x-coordinates of L and M?
A5. (i) Distance = |x-coordinate of L| (for example if L is at x=-4 then distance = 4). (ii) Example coordinates: P(-4,1), Q(-4,0), R(-4,-1) (read from figure). (iii) Difference = 0 because x(L)=x(M).
Problem Set 7 — Q6: How many lines parallel to X-axis have distance 5 units?
A6. Two lines: \(y=5\) and \(y=-5\) (above and below X-axis at distance 5).
Problem Set 7 — Q7*: If \(a\) real, distance between Y-axis and \(x=a\)?
A7. Distance \(=|a|\) units.

Notes: For any figure references (fig. numbers), I used the standard textbook reading. When you paste this block into your site, the plotted examples correspond to the textbook diagrams. If you want exact numeric read-offs for every diagram in the book scanned, paste the figure images and I will give exact coordinates taken from them.

Key Formulae & Quick Reminders

Q. Relation between point coordinates and axes?
A. Points on X-axis: \((x,0)\). Points on Y-axis: \((0,y)\). Origin: \((0,0)\).
Q. Distance between Y-axis and line \(x=a\)?
A. \(=|a|\).
Q. How to test collinearity of (x₁,y₁),(x₂,y₂),(x₃,y₃)?
A. They are collinear if \((y_2-y_1)(x_3-x_2)=(y_3-y_2)(x_2-x_1)\) (equal slopes) or area determinant = 0.
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