7.1 Coordinate geometry, equation of tangent, transformations  (Page 2/2)

The tangent (line $g$ ) is perpendicular to this line. Therefore,

So,

Now, we know that the tangent passes through $(x_1,y_1)$ so the equation is given by:

For example, find the equation of the tangent to the circle at point $(1,1)$ . The centre of the circle is at $(0,0)$ . The equation of the circle is $x^2+y^2=2$ .

Use

with $(x_0,y_0)=(0,0)$ and $(x_1,y_1)=(1,1)$ .

Co-ordinate geometry

1. Find the equation of the cicle:
   1. with centre $(0;5)$ and radius 5
   2. with centre $(2;0)$ and radius 4
   3. with centre $(5;7)$ and radius 18
   4. with centre $(-2;0)$ and radius 6
   5. with centre $(-5;-3)$ and radius $\sqrt{3}$
2. 1. Find the equation of the circle with centre $(2;1)$ which passes through $(4;1)$ .
   2. Where does it cut the line $y=x+1$ ?
   3. Draw a sketch to illustrate your answers.
3. 1. Find the equation of the circle with center $(-3;-2)$ which passes through $(1;-4)$ .
   2. Find the equation of the circle with center $(3;1)$ which passes through $(2;5)$ .
   3. Find the point where these two circles cut each other.
4. Find the center and radius of the following circles:
   1. ${(x-9)}^2+{(y-6)}^2=36$
   2. ${(x-2)}^2+{(y-9)}^2=1$
   3. ${(x+5)}^2+{(y+7)}^2=12$
   4. ${(x+4)}^2+{(y+4)}^2=23$
   5. $3{(x-2)}^2+3{(y+3)}^2=12$
   6. $x^2-3x+9=y^2+5y+25=17$
5. Find the $x-$ and $y-$ intercepts of the following graphs and draw a sketch to illustrate your answer:
   1. ${(x+7)}^2+{(y-2)}^2=8$
   2. $x^2+{(y-6)}^2=100$
   3. ${(x+4)}^2+y^2=16$
   4. ${(x-5)}^2+{(y+1)}^2=25$
6. Find the center and radius of the following circles:
   1. $x^2+6x+y^2-12y=-20$
   2. $x^2+4x+y^2-8y=0$
   3. $x^2+y^2+8y=7$
   4. $x^2-6x+y^2=16$
   5. $x^2-5x+y^2+3y=-\frac{3}{4}$
   6. $x^2-6nx+y^2+10ny=9n^2$
7. Find the equations to the tangent to the circle:
   1. $x^2+y^2=17$ at the point $(1;4)$
   2. $x^2+y^2=25$ at the point $(3;4)$
   3. ${(x+1)}^2+{(y-2)}^2=25$ at the point $(3;5)$
   4. ${(x-2)}^2+{(y-1)}^2=13$ at the point $(5;3)$

Transformations

Rotation of a point about an angle $\theta$

First we will find a formula for the co-ordinates of $P$ after a rotation of $\theta$ .

We need to know something about polar co-ordinates and compound angles before we start.

Polar co-ordinates

1. $P(x;y)$ rectangular co-ordinates
2. or $P(rcos\alpha ;rsin\alpha )$ polar co-ordinates.

Compound angles

(See derivation of formulae in Ch. 12)

Now consider $P^{\text{'}}$ After a rotation of $\theta$

which gives the formula $P^{\text{'}}=\left[(,x,cos,\theta ,-,y,sin,\theta ,;,y,cos,\theta ,+,x,sin,\theta ,)\right]$ .

So to find the co-ordinates of $P(1;\sqrt{3})$ after a rotation of 45 ${}^{\circ }$ , we arrive at:

Rotations

Any line $OP$ is drawn (not necessarily in the first quadrant), making an angle of $\theta$ degrees with the $x$ -axis. Using the co-ordinates of $P$ and the angle $\alpha$ , calculate the co-ordinates of $P^{\text{'}}$ , if the line $OP$ is rotated about the origin through $\alpha$ degrees.

Characteristics of transformations

Rigid transformations like translations, reflections, rotations and glide reflections preserve shape and size, and that enlargement preserves shape but not size.

Geometric transformations:

Draw a large 15 $\times$ 15 grid and plot $▵ABC$ with $A(2;6)$ , $B(5;6)$ and $C(5;1)$ . Fill in the lines $y=x$ and $y=-x$ . Complete the table below , by drawing the images of $▵ABC$ under the given transformations. The first one has been done for you.

A transformation that leaves lengths and angles unchanged is called a rigid transformation. Which of the above transformations are rigid?

Exercises

1. $\Delta ABC$ undergoes several transformations forming $\Delta A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ . Describe the relationship between the angles and sides of $\Delta ABC$ and $\Delta A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ (e.g., they are twice as large, the same, etc.)

   Transformation	Sides	Angles	Area
   Reflect
   Reduce by a scale factor of 3
   Rotate by 90 ${}^{\circ }$
   Translate 4 units right
   Enlarge by a scale factor of 2

2. $\Delta DEF$ has $\widehat{E}={30}^{\circ }$ , $DE=4\mathrm{cm}$ , $EF=5\mathrm{cm}$ . $\Delta DEF$ is enlarged by a scale factor of 6 to form $\Delta D^{\text{'}}{E}^{\text{'}}{F}^{\text{'}}$ .
   1. Solve $\Delta DEF$
   2. Hence, solve $\Delta D^{\text{'}}{E}^{\text{'}}{F}^{\text{'}}$
3. $\Delta XYZ$ has an area of $6{\mathrm{cm}}^2$ . Find the area of $\Delta X^{\text{'}}{Y}^{\text{'}}{Z}^{\text{'}}$ if the points have been transformed as follows:
   1. $(x,y)\to (x+2;y+3)$
   2. $(x,y)\to (y;x)$
   3. $(x,y)\to (4x;y)$
   4. $(x,y)\to (3x;y+2)$
   5. $(x,y)\to (-x;-y)$
   6. $(x,y)\to (x;-y+3)$
   7. $(x,y)\to (4x;4y)$
   8. $(x,y)\to (-3x;4y)$