Circle $C_1$ has equation $x^2 + y^2 = 100$ Circle $C_2$ has equation $(x - 15)^2 + y^2 = 40$ The circles meet at points $A$ and $B$ as shown in Figure 3. (a) Show that angle $AOB = 0.635$ radians to 3 significant figures, where $O$ is the origin. (b) The region shown shaded in Figure 3 is bounded by $C_1$ and $C_2$. Give the perimeter of the shaded region, giving your answer to one decimal place.

A-Level Edexcel Maths Pure Trigonometric Equations: Circle $C

Circle $C_1$ has equation $x^2 + y^2 = 100$ Circle $C_2$ has equation $(x - 15)^2 + y^2 = 40$ The circles meet at points $A$ and $B$ as shown in Figure 3 - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 1

Question 12

Circle-$C_1$-has-equation-$x^2-+-y^2-=-100$---Circle-$C_2$-has-equation-$(x---15)^2-+-y^2-=-40$---The-circles-meet-at-points-$A$-and-$B$-as-shown-in-Figure-3-Edexcel-A-Level Maths Pure-Question 12-2020-Paper 1.png

Circle $C_1$ has equation $x^2 + y^2 = 100$ Circle $C_2$ has equation $(x - 15)^2 + y^2 = 40$ The circles meet at points $A$ and $B$ as shown in Figure 3. (... show full transcript

Worked Solution & Example Answer:Circle $C_1$ has equation $x^2 + y^2 = 100$ Circle $C_2$ has equation $(x - 15)^2 + y^2 = 40$ The circles meet at points $A$ and $B$ as shown in Figure 3 - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 1

Step 1

Show that angle AOB = 0.635 radians to 3 significant figures

96%

114 rated

Answer

To find angle $AOB$ , begin by determining the coordinates of the intersection points of the circles $C_1$ and $C_2$ .

Find Coordinates of Points A and B:
- Solve the system of equations:
  $x^2 + y^2 = 100 ag{1}$ $(x - 15)^2 + y^2 = 40 ag{2}$
- Expanding (2):
  $x^2 - 30x + 225 + y^2 = 40$
- Substitute (1) into this to find:
  $100 - 30x + 225 = 40$
- This simplifies to:
  $-30x + 325 = 40 \ -30x = -285 \ x = 9.5$
- Substitute $x = 9.5$ back into (1) to find $y$ :
  $(9.5)^2 + y^2 = 100 \ 90.25 + y^2 = 100 \ y^2 = 9.75 \ y = rac{ ext{sqrt}(39)}{2}$
Using Trigonometry to Find Angle AOB:
- In triangle $AOB$ , using the coordinates for points $A$ and $B$ with center $O(0, 0)$ :
- Use cosine rule or derive directly using tangent:
  $ext{cos} heta = rac{OA^2 + OB^2 - AB^2}{2 imes OA imes OB}$
- Solve for the angle in radians:
  $heta = 2 ext{arccos}igg( rac{9.5}{10}igg) = 0.635$
- Angle $AOB$ is thus confirmed as $0.635$ radians.

Step 2

Find the perimeter of the shaded region

99%

104 rated

Answer

To find the perimeter of the shaded region bounded by circles $C_1$ and $C_2$ , follow these steps:

Determine the Radii:
- Circle $C_1$ : Radius $r_1 = 10$ (from $x^2 + y^2 = 100$ )
- Circle $C_2$ : Radius $r_2 = ext{sqrt}(40) = 6.32$ (from $(x - 15)^2 + y^2 = 40$ )
Calculate the Length of Arc OA and Arc OB:
- The angle $AOB = 0.635$ radians, so the lengths are:
  $L_1 = r_1 heta = 10 imes 0.635 = 6.35$ $L_2 = r_2 heta = 6.32 imes 0.635 = 4.01$
Perimeter Calculation:
- The total perimeter is the sum of arcs OA and OB:
  $P = L_1 + L_2 = 6.35 + 4.01 = 10.36$
- Round to one decimal place gives:
  $P ext{ (shaded perimeter)} = 10.4$

Thus, the perimeter of the shaded region is approximately 10.4 units.

Circle $C_1$ has equation $x^2 + y^2 = 100$ Circle $C_2$ has equation $(x - 15)^2 + y^2 = 40$ The circles meet at points $A$ and $B$ as shown in Figure 3 - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 1

Worked Solution & Example Answer:Circle $C_1$ has equation $x^2 + y^2 = 100$ Circle $C_2$ has equation $(x - 15)^2 + y^2 = 40$ The circles meet at points $A$ and $B$ as shown in Figure 3 - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 1

Show that angle AOB = 0.635 radians to 3 significant figures

Find the perimeter of the shaded region

Join the A-Level students using SimpleStudy...