Coordinate Geometry

Chapter 4 · Mathematics · 6 questions in the paper · 6 min read

Coordinate Geometry

What the syllabus asks for 4
  • Straight lines, pair of lines
  • Circles, equations of circle in different forms, tangent and normal
  • Conic sections: parabola, ellipse and hyperbola, standard equations and simple properties
  • Coordinates in space, plane and its equation

Coordinate geometry carries about 6 questions (our estimate), and the official booklet's own model question — "the value of k for which x+y-1=0 touches y^2=kx" — shows the style: a tangency condition, a standard property of a conic, or a plane in space. Memorising the standard forms and the three tangency conditions (c^2=r^2(1+m^2), c=a/m, c^2=a^2m^2\pm b^2) answers more than half of them.

What the paper asks from this chapter
  • Distance of a point from a line, angle between two lines, concurrency, area of a triangle.
  • Pair of lines ax^2+2hxy+by^2=0: angle \tan\theta=\frac{2\sqrt{h^2-ab}}{|a+b|}; perpendicular iff a+b=0.
  • Circle: centre and radius from the general equation, tangent, length of tangent \sqrt{S_1}, radical axis.
  • Conics: focus, directrix, latus rectum, eccentricity; tangent in slope form.
  • Space: distance, direction cosines, plane equation, distance from a point to a plane.

4.1 Straight lines, pair of lines

Form Equation
slope-intercept y=mx+c
point-slope y-y_1=m(x-x_1)
two-point \dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}
intercept \dfrac{x}{a}+\dfrac{y}{b}=1
normal x\cos\alpha+y\sin\alpha=p
general Ax+By+C=0, slope -A/B
  • Distance of (x_1,y_1) from Ax+By+C=0: \dfrac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}. Between parallel lines Ax+By+C_1=0, Ax+By+C_2=0: \dfrac{|C_1-C_2|}{\sqrt{A^2+B^2}}.
  • Angle between lines of slopes m_1,m_2: \tan\theta=\Big|\dfrac{m_1-m_2}{1+m_1m_2}\Big|; parallel m_1=m_2, perpendicular m_1m_2=-1.
  • Family of lines through the intersection of L_1=0,L_2=0: L_1+\lambda L_2=0. Three lines are concurrent if the determinant of their coefficients is 0.
  • Area of the triangle (x_i,y_i): \tfrac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|.
  • Bisectors of the angles between two lines: \dfrac{A_1x+B_1y+C_1}{\sqrt{A_1^2+B_1^2}}=\pm\dfrac{A_2x+B_2y+C_2}{\sqrt{A_2^2+B_2^2}}.
  • Foot of the perpendicular and image: solve using the normal direction (A,B).

Pair of lines

ax^2+2hxy+by^2=0 (homogeneous) is a pair of lines through the origin with m_1+m_2=-\frac{2h}{b}, m_1m_2=\frac{a}{b}. Angle: \tan\theta=\dfrac{2\sqrt{h^2-ab}}{a+b}. Real and distinct if h^2>ab, coincident if h^2=ab; perpendicular if a+b=0. The general second-degree equation ax^2+2hxy+by^2+2gx+2fy+c=0 is a pair of lines iff abc+2fgh-af^2-bg^2-ch^2=0.

4.2 Circles, equations in different forms, tangent and normal

  • Centre (h,k), radius r: (x-h)^2+(y-k)^2=r^2.
  • General: x^2+y^2+2gx+2fy+c=0, centre (-g,-f), radius \sqrt{g^2+f^2-c}.
  • Diameter form: ends (x_1,y_1),(x_2,y_2): (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0.
  • Parametric: (h+r\cos\theta,\,k+r\sin\theta).
  • Through three points: substitute into the general form and solve.

Tangent at (x_1,y_1): xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0 (for x^2+y^2=r^2: xx_1+yy_1=r^2). The normal passes through the centre. Line y=mx+c touches x^2+y^2=r^2 iff c^2=r^2(1+m^2). Length of tangent from P: \sqrt{S_1} where S_1 is the circle's expression at P. Chord at distance d from the centre: 2\sqrt{r^2-d^2}. Two circles: radical axis S_1-S_2=0; orthogonal iff 2g_1g_2+2f_1f_2=c_1+c_2; with centres distance d apart: externally tangent if d=r_1+r_2, internally if d=|r_1-r_2|.

4.3 Conic sections: parabola, ellipse and hyperbola

All conics are \dfrac{\text{distance from focus}}{\text{distance from directrix}}=e.

Parabola y^2=4ax Ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a>b) Hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1
e 1 \sqrt{1-b^2/a^2}<1 \sqrt{1+b^2/a^2}>1
focus (a,0) (\pm ae,0) (\pm ae,0)
directrix x=-a x=\pm a/e x=\pm a/e
latus rectum 4a 2b^2/a 2b^2/a
tangent (slope m) y=mx+a/m y=mx\pm\sqrt{a^2m^2+b^2} y=mx\pm\sqrt{a^2m^2-b^2}
tangent at (x_1,y_1) yy_1=2a(x+x_1) \frac{xx_1}{a^2}+\frac{yy_1}{b^2}=1 \frac{xx_1}{a^2}-\frac{yy_1}{b^2}=1
parametric (at^2,2at) (a\cos\theta,b\sin\theta) (a\sec\theta,b\tan\theta)
other vertex origin major axis 2a, minor 2b; sum of focal distances 2a asymptotes y=\pm\frac{b}{a}x; |PF_1-PF_2|=2a

Other orientations: x^2=4ay opens upward with focus (0,a); y^2=-4ax opens left. For a hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=-1 the transverse axis is vertical. Rectangular hyperbola xy=c^2 (a=b, e=\sqrt{2}); asymptotes are the axes. The focal distance of (x_1,y_1) on the parabola is x_1+a. The parabola y=px^2+qx+r has vertex at x=-\frac{q}{2p}.

The booklet's tangent question

Problem. For what k is x+y-1=0 tangent to y^2=kx?

Solution. y^2=4ax with a=k/4. The line is y=-x+1, m=-1, c=1. Condition c=a/m: 1=\dfrac{k/4}{-1}, so k=-4.

4.4 Coordinates in space, plane and its equation

  • Distance \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}. Section formula (ratio m:n): \Big(\dfrac{mx_2+nx_1}{m+n},\dots\Big).
  • Direction cosines l,m,n with l^2+m^2+n^2=1; direction ratios a,b,c give l=\dfrac{a}{\sqrt{a^2+b^2+c^2}}. A line equally inclined to the axes has l=m=n=\pm\frac{1}{\sqrt{3}}.
  • Angle between lines \cos\theta=|l_1l_2+m_1m_2+n_1n_2|.
  • Plane ax+by+cz+d=0 with normal (a,b,c); intercept form \frac{x}{p}+\frac{y}{q}+\frac{z}{r}=1; through three points: determinant condition; through (x_1,y_1,z_1) with normal (a,b,c): a(x-x_1)+b(y-y_1)+c(z-z_1)=0.
  • Distance of (x_1,y_1,z_1) from the plane: \dfrac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}. Between parallel planes: \dfrac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}.
  • Angle between planes = angle between normals. Plane and line: \sin\theta=\dfrac{|aL+bM+cN|}{\sqrt{a^2+b^2+c^2}\sqrt{L^2+M^2+N^2}}.
  • Line through (x_1,y_1,z_1) with direction ratios a,b,c: \dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}.

Formula and fact sheet

Topic Result
point–line distance \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}
angle between lines \tan\theta=|\frac{m_1-m_2}{1+m_1m_2}|
circle tangent c^2=r^2(1+m^2)
tangent length \sqrt{S_1}
parabola y=mx+a/m, LR =4a
ellipse / hyperbola e \sqrt{1-b^2/a^2} / \sqrt{1+b^2/a^2}
plane distance \frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}

Chapter summary

  • Distance, angle and area formulas for lines are one-liners; learn them exactly.
  • x^2+y^2+2gx+2fy+c=0: centre (-g,-f), radius \sqrt{g^2+f^2-c}.
  • Tangency to a circle is c^2=r^2(1+m^2); to a parabola c=a/m.
  • Latus rectum: 4a for the parabola, 2b^2/a for ellipse and hyperbola.
  • A plane's normal is (a,b,c); parallel planes differ only in d.

Practice questions

Set A — Recall (12 questions)

  1. The slope of the line 3x+4y=12 isa3/4b-3/4c4/3d-4/3
  2. The distance of the origin from the line 3x+4y-10=0 isa2b10c5d2/5
  3. The acute angle between the lines y=x and y=\sqrt{3}x isa30^\circb45^\circc15^\circd60^\circ
  4. The centre and radius of x^2+y^2-6x+8y-11=0 area(-3,4), 6b(3,-4), 36c(3,-4), 5d(3,-4), 6
  5. The length of the tangent from (5,0) to x^2+y^2=9 isa3b4c5d16
  6. The focus of the parabola y^2=8x isa(4,0)b(8,0)c(2,0)d(0,2)
  7. The length of the latus rectum of \frac{x^2}{25}+\frac{y^2}{9}=1 isa9/5b6c10d18/5
  8. The eccentricity of \frac{x^2}{16}-\frac{y^2}{9}=1 isa5/4b4/5c7/4d3/4
  9. The distance between (1,2,3) and (4,6,15) isa12b13c11d14
  10. The distance of (1,2,3) from the plane x+2y+2z=3 isa2b11/3c3d8/3
  11. ax^2+2hxy+by^2=0 represents perpendicular lines ifah=0bab=h^2ca+b=0da=b
  12. The radius of the circle with the ends of a diameter at (1,2) and (5,6) isa2\sqrt{2}b4c8d4\sqrt{2}
Answers and explanations — Set A

1. (b) — y=-\frac{3}{4}x+3.

2. (a) — \frac{|-10|}{\sqrt{9+16}}=2.

3. (c) — slopes 1 and \sqrt{3}: \tan\theta=\frac{\sqrt{3}-1}{\sqrt{3}+1}=2-\sqrt{3}=\tan15^\circ (also 60^\circ-45^\circ).

4. (d) — g=-3, f=4, c=-11: centre (3,-4), r=\sqrt{9+16+11}=6.

5. (b) — \sqrt{S_1}=\sqrt{25-9}=4.

6. (c) — 4a=8, a=2.

7. (d) — \frac{2b^2}{a}=\frac{18}{5}.

8. (a) — e=\sqrt{1+\frac{9}{16}}=\frac{5}{4}.

9. (b) — \sqrt{9+16+144}=13.

10. (d) — \frac{|1+4+6-3|}{\sqrt{1+4+4}}=\frac{8}{3}.

11. (c) — a+b=0 (coefficient sum of x^2,y^2).

12. (a) — diameter =\sqrt{16+16}=4\sqrt{2}, radius 2\sqrt{2}.

Set B — Understanding (8 questions)

  1. The line through (1,2) perpendicular to 2x+3y=5 isa3x-2y+1=0b3x+2y-7=0c2x+3y-8=0d2x-3y+4=0
  2. The distance between the parallel lines x+2y-3=0 and x+2y+7=0 isa10b2\sqrt{5}c4/\sqrt{5}d\sqrt{5}
  3. The lines x+y=2, 2x-y=1 and 3x+ky=7 are concurrent for k=a2b3c4d5
  4. The tangent to x^2+y^2=25 at (3,4) isa4x+3y=25b3x-4y=25c3x+4y=5d3x+4y=25
  5. The line y=x+c touches the circle x^2+y^2=8 forac=\pm2bc=\pm\sqrt{8}cc=\pm4dc=\pm8
  6. The parabola with focus (3,0) and directrix x=-3 isay^2=12xby^2=6xcy^2=3xdx^2=12y
  7. An ellipse has foci (\pm3,0) and minor axis of length 8. Its eccentricity isa4/5b3/5c3/4d5/3
  8. The plane through (1,0,0), (0,2,0) and (0,0,3) isax+2y+3z=6b3x+2y+z=6c2x+3y+6z=6d6x+3y+2z=6
Answers and explanations — Set B

1. (a) — slope of the given line is -\frac{2}{3}, so the perpendicular's is \frac{3}{2}: y-2=\frac{3}{2}(x-1)\Rightarrow3x-2y+1=0.

2. (b) — \frac{|-3-7|}{\sqrt{5}}=\frac{10}{\sqrt{5}}=2\sqrt{5}.

3. (c) — the first two meet at (1,1); then 3+k=7.

4. (d) — xx_1+yy_1=r^2: 3x+4y=25.

5. (c) — c^2=r^2(1+m^2)=8\times2=16.

6. (a) — a=3, y^2=4ax=12x.

7. (b) — b=4, ae=3: a^2=16+9=25, a=5, e=\frac{3}{5}.

8. (d) — intercept form x+\frac{y}{2}+\frac{z}{3}=1; multiply by 6.

Set C — Application (6 questions)

  1. The area of the triangle with vertices (1,1), (4,5), (7,2) isa21b10.5c7d12
  2. The circle through (0,0), (4,0) and (0,6) has radius^2 equal toa5b25c13d10
  3. The radical axis of x^2+y^2=9 and x^2+y^2-4x-6y+4=0 isa4x+6y-13=0b4x+6y+13=0c2x+3y-13=0d4x-6y-13=0
  4. The tangent to y^2=16x with slope 1 isax-y-4=0bx-y+16=0cx+y+4=0dx-y+4=0
  5. The tangent of the acute angle between the lines x^2-5xy+4y^2=0 isa5/3b3/5c3/4d4/5
  6. The plane through (1,2,3) parallel to x+2y+3z=7 isa3x+2y+z=10bx+2y+3z=7cx+2y+3z=14dx+2y+3z=6
Answers and explanations — Set C

1. (b) — \frac{1}{2}|1(5-2)+4(2-1)+7(1-5)|=\frac{1}{2}|3+4-28|=10.5.

2. (c) — a right angle at the origin, so the hypotenuse (4,0)–(0,6) is a diameter: centre (2,3), r^2=4+9=13.

3. (a) — S_1-S_2=(x^2+y^2-9)-(x^2+y^2-4x-6y+4)=4x+6y-13.

4. (d) — a=4, m=1: y=x+\frac{a}{m}=x+4.

5. (b) — h=-\frac{5}{2}, a=1, b=4: \frac{2\sqrt{25/4-4}}{5}=\frac{2\cdot1.5}{5}=\frac{3}{5} (factors (x-y)(x-4y), slopes 1,\frac{1}{4}).

6. (c) — same normal (1,2,3); 1+4+9=14.