Secondary 5 Analytic Geometry Worksheet

All 16 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Analytic Geometry

   Analytic geometry describes a geometric object by an equation on the Cartesian plane. For each equation below, name the curve it represents (line, circle, ellipse, parabola or hyperbola) and give the one feature of the equation that settles your choice.

   1. $x^2+y^2-6x+4y=12$
   2. ${\displaystyle \frac{(x+1{)}^2}{9}}-{\displaystyle \frac{(y-2{)}^2}{4}}=1$
   3. $y=-2(x-3{)}^2+5$
   4. $3x-5y=15$
   5. ${\displaystyle \frac{x^2}{25}}+{\displaystyle \frac{y^2}{16}}=1$
2. Q2Circle (Conic)

   A city park has a circular splash pad whose centre, on the park map, is the point $(-2,5)$ and whose radius is $6$~m.

   1. Write the equation of the edge of the splash pad.
   2. A drinking fountain stands at $(3,7)$. Is it inside the splash pad, on its edge, or outside? Justify with a computation.
3. Q3Conics

   Write $9x^2+4y^2-54x+8y+49=0$ in standard form. Name the conic, and give its centre and the length of each of its axes.

4. Q4Dilations on a Cartesian Plane

   Triangle $ABC$ has vertices $A(-1,2)$, $B(4,0)$ and $C(2,-3)$. It undergoes the dilation $h_{(O,3)}$ centred at the origin with ratio $3$.

   1. Give the coordinates of $A^{\prime }$, $B^{\prime }$ and $C^{\prime }$.
   2. By what factor is the area of the triangle multiplied?
5. Q5Ellipse (Conic)

   A trampoline frame is an ellipse $10$~m long and $6$~m wide, centred at the origin with its longest measurement along the $x$-axis.

   1. Write its equation.
   2. Give the coordinates of its vertices and of its foci.
6. Q6Geometric Transformations in the Cartesian Plane

   For each mapping rule, name the geometric transformation completely (including its centre, axis or arrow as appropriate) and state whether it is an isometry.

   1. $(x,y)\mapsto (x-3,y+5)$
   2. $(x,y)\mapsto (-x,y)$
   3. $(x,y)\mapsto (3x,3y)$
   4. $(x,y)\mapsto (y,-x)$
7. Q7Hyperbola (Conic)

   Consider ${\displaystyle \frac{(x-1{)}^2}{9}}-{\displaystyle \frac{(y+2{)}^2}{16}}=1$. Give

   1. its centre and its vertices;
   2. its foci;
   3. the equations of its asymptotes.
8. Q8Parabola (Conic)

   A parabola has its vertex at the origin and its focus at $(0,3)$.

   1. Write its equation and the equation of its directrix.
   2. How wide is the parabola along the horizontal line through the focus?
9. Q9Points of Intersection Between a Line and a Conic

   Find the coordinates of the points where the line $y=x+1$ meets the circle $x^2+y^2=25$.

10. Q10Points of Intersection Between a Parabola and a Conic

    Find all points of intersection of the parabola $y=x^2-2$ and the circle $x^2+y^2=4$.

11. Q11Reflections on a Cartesian Plane

    Triangle $ABC$ has vertices $A(-3,1)$, $B(2,4)$ and $C(5,-2)$. Give the coordinates of its image under

    1. the reflection in the $x$-axis;
    2. the reflection in the line $y=x$.
12. Q12Rotations on a Cartesian Plane

    Give the image of the point $A(5,-2)$ under each rotation centred at the origin.

    1. $r_{(O,{90}^{\circ })}$
    2. $r_{(O,{180}^{\circ })}$
    3. $r_{(O,-{90}^{\circ })}$
13. Q13The Composition of Geometric Transformations in a Cartesian Plane

    Let $s$ be the reflection in the $x$-axis and $t$ the translation $t_{(3,-2)}$.

    1. Give the mapping rule of the composition $t\circ s$ (first $s$, then $t$).
    2. Find the image of $A(-2,5)$ under $t\circ s$.
    3. Is $t\circ s$ an isometry? Justify.
14. Q14The Division Point and Midpoint of a Segment

    A hiking trail runs in a straight line from the trailhead $A(-6,4)$ to the lookout $B(4,-6)$ (units in hundreds of metres).

    1. Find the midpoint $M$ of $\overline{AB}$.
    2. Find the point $P$ that divides $\overline{AB}$ in the ratio $2:3$ starting from $A$.
15. Q15The Equations of Parallel or Perpendicular Lines

    Consider the line $d_1:2x+5y=20$.

    1. Find the equation of the line perpendicular to $d_1$ passing through $(6,-1)$.
    2. Find the equation of the line parallel to $d_1$ passing through $(0,3)$.
16. Q16Translations on a Cartesian Plane

    A design is moved by the translation $t_{(-5,3)}$.

    1. Find the image of the point $A(7,-2)$.
    2. A circular hole in the design has equation $(x-2{)}^2+(y+1{)}^2=9$. Give the equation of its image, and state the image's centre and radius.

The 17 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Naming a conic from its equation, in four seconds

Q1 gives five equations and asks which curve each one is, plus the single feature that settles it. That "single feature" phrasing is deliberate: you are not meant to complete the square five times. You are meant to look at the squared terms.

Completing the square is the engine of this whole topic

Q2 and Q3 both live or die on it, and so does part of Q13's family of questions. An equation given in general form tells you almost nothing; the same equation in standard form hands you the centre, the radius or the axes by inspection. On my own example:

Group the x terms and the y terms, halve each linear coefficient, square it, and add it to both sides:

Q2 also asks whether a given point is inside, on, or outside a circle. Two equally valid methods: compute the distance from the centre and compare it with the radius, or substitute the point into the left side of the standard-form equation and compare the result with r². Smaller means inside, equal means on the edge, larger means outside. The second method avoids a square root entirely.

a, b and c: the one relation that flips between ellipse and hyperbola

Q5 and Q7 both ask for foci, and they use opposite formulas. This is the fact most worth having cold:

For an ellipse, a is always the larger of the two, the major axis is the one it belongs to, and the foci sit on that major axis, inside the curve. A trampoline 10 m long and 6 m wide has a = 5 and b = 3 — half-lengths, not full lengths, which is the trap in Q5: the equation wants semi-axes while the situation gives you widths.

For a hyperbola, a is the denominator of the positive term whether or not it is larger, the transverse axis follows that variable, the vertices are a from the centre along it, and the foci are c from the centre along the same line — always further out than the vertices, since c > a.

The asymptotes in Q7(c) come straight from the standard form: replace the 1 on the right by 0 and solve. For a centre at (h, k) and a horizontal transverse axis that gives

with the fraction inverted when the transverse axis is vertical. Give them as two separate equations of lines unless the question accepts the ± form, and remember the curve approaches them without ever touching — a point lying exactly on an asymptote cannot be on the hyperbola.

The parabola as a conic: 4c does three jobs

Q8 treats the parabola the conic way — by its focus and directrix rather than its vertex form. With the vertex at the origin and an upward opening:

The same c is the distance from the vertex up to the focus and down to the directrix, and 4c is the width of the curve measured along the horizontal line through the focus. So a single given — a focus, a directrix, or one rim point — determines everything else. If the parabola opens sideways, the roles of x and y swap: y² = 4cx.

Intersections: substitute, and be ready to reject a root

Q9 crosses a line with a circle, Q10 crosses a parabola with a circle. Both are solved by substitution, but the substitution to choose is different, and choosing well is most of the work.

Line and conic (Q9): the line gives y in terms of x. Put that expression into the conic, expand, and you are left with a quadratic in x. Solve it, then feed each root back through the line to get its y — the line is the simpler of the two rules and cannot introduce a false partner. The number of real roots is the number of intersection points: two, one (tangent), or none.

Parabola and circle (Q10): do not substitute y = x² − 2 into the circle — that produces a fourth-degree equation. Substitute the other way: read x² off the parabola and replace x² in the circle. The system collapses to a quadratic in y.

The transformations, as point rules

Q4, Q6, Q11, Q12, Q13 and Q16 all reduce to a handful of mappings. Learn them as what they do to (x, y), not as sentences:

Translations, reflections and rotations preserve every distance — they are isometries, which is what Q6 and Q13 are asking you to decide. A dilation with k ≠ ±1 is not: it multiplies every length by |k| and every area by k², which is the whole of Q4(b). It is still a similarity, so angles survive unchanged.

When the centre of the dilation or rotation is not the origin, work with vectors from the centre. For a dilation of ratio k centred at C, the image of P is the point with CP′ = k · CP; a negative k puts the image on the opposite side of C. For a rotation, the same subtraction gets you to a rotation about the origin, which you then undo by adding C back.

A composition such as Q13's t ∘ s is read right to left: s acts first. Substitute the first rule into the second and simplify to a single mapping. Composition of transformations is not commutative — swapping the order generally moves the image — and a composition of isometries is again an isometry, which answers the "is it an isometry?" part without any computation.

Transforming a curve: the substitution goes the other way

Q16 applies a translation to a circle rather than to a point, and this is where almost everyone signs the wrong way. To translate a curve by (a, b), replace x by x − a and y by y − b in its equation.

Midpoint, division point, parallel and perpendicular

Q14 asks for a midpoint and then for a point dividing a segment in a given ratio. The midpoint is the average of the coordinates. The division point in the ratio r : s starting from A is

with AB the vector from A to B. Note the denominator: a ratio of 2 : 3 means two parts out of five, not out of three. The starting endpoint matters too — dividing from B instead of A in the same ratio lands somewhere else entirely.

Q15 needs one fact and one formula. From a line written Ax + By = C, the slope is −A ⁄ B; parallel lines share it, and perpendicular lines have slopes whose product is −1, so you take the negative reciprocal. Then use the point-slope form y − y₁ = m(x − x₁) and simplify. The one case the rule does not cover is a vertical line, whose slope does not exist and whose perpendiculars are horizontal — worth a sentence if it ever comes up.

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Getting the most out of it

Draw it, even badly

A rough sketch with the centre marked and the axes labelled catches sign errors that algebra hides. On the intersection questions it also tells you in advance how many points to expect, so an answer with the wrong count is caught immediately.

Put every conic in standard form before answering anything

Centre, semi-axes, foci and asymptotes are all read off the standard form and none of them are visible in the general form. Make it a reflex: complete the square first, answer the question second.

Drill the transformation rules as point-images

Write the ten mappings above on a card and test yourself by applying each to a single point such as (3, −1). Recognising a rule from its effect is exactly what Q6 asks for, and it is recall, not reasoning.

Common questions