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Motion In A Plane

Question
CBSEENPH11018371
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Point C is situated on the circumference of a circle of diameter AB. Show that triangle ACB is right angled triangle.

Solution

Let a be the radius of circle. 
Here, OB with rightwards harpoon with barb upwards on top space equals space P with rightwards harpoon with barb upwards on top space a n d space stack O C with rightwards harpoon with barb upwards on top space equals space Q with rightwards harpoon with barb upwards on top
therefore space OA with rightwards harpoon with barb upwards on top space equals space minus straight P with rightwards harpoon with barb upwards on top 
Also, open vertical bar OB with rightwards harpoon with barb upwards on top close vertical bar space equals space open vertical bar stack O C with rightwards harpoon with barb upwards on top close vertical bar equals open vertical bar stack O A with rightwards harpoon with barb upwards on top close vertical bar space equals space a 
Now, from increment AOC comma space we space have 
space space space space space space space space space space space space space space space space space space space space stack O C with rightwards harpoon with barb upwards on top space plus space stack C A with rightwards harpoon with barb upwards on top space equals space stack O A with rightwards harpoon with barb upwards on top rightwards double arrow space space space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top space equals space stack O A with rightwards harpoon with barb upwards on top space minus space stack O C with rightwards harpoon with barb upwards on top space equals space minus P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top space  
In increment BOC comma space
space space space space space space space space space space space space space OC with rightwards harpoon with barb upwards on top space plus space stack C B with rightwards harpoon with barb upwards on top space equals space stack O B with rightwards harpoon with barb upwards on top rightwards double arrow space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top space equals space stack O B with rightwards harpoon with barb upwards on top space minus space stack O C with rightwards harpoon with barb upwards on top space equals negative P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top N o w comma space space space space space open parentheses stack C A with rightwards harpoon with barb upwards on top close parentheses. open parentheses stack C B with rightwards harpoon with barb upwards on top close parentheses space equals space open parentheses negative P with rightwards harpoon with barb upwards on top minus Q with rightwards harpoon with barb upwards on top close parentheses. open parentheses P with rightwards harpoon with barb upwards on top minus Q with rightwards harpoon with barb upwards on top close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus P with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top space minus space Q with rightwards harpoon with barb upwards on top. P with rightwards harpoon with barb upwards on top space plus space P with rightwards harpoon with barb upwards on top. Q with rightwards harpoon with barb upwards on top space plus space Q with rightwards harpoon with barb upwards on top. Q with rightwards harpoon with barb upwards on top space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus P squared space plus space Q squared space equals space minus a squared space plus space a squared space equals space 0 space rightwards double arrow space space space space space space space space space space space space space space stack C A with rightwards harpoon with barb upwards on top perpendicular stack C B with rightwards harpoon with barb upwards on top  
Therefore, triangle ACB is a right angled triangle. 

Some More Questions From Motion in A Plane Chapter

What is a vector quantity?

Give three examples of scalar quantities.

Give three examples of vector quantities.

What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

Is a physical quantity, having a direction, necessarily a vector?

Name a quantity which is both scalar as well as vector.

Which of the followings is/are vector quantities: work, power, acceleration, mass, momentum?

Pick up the odd one from the following: force, linear momentum, mass, impulse.

Pick out the only vector quantities in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction and charge.

Pick out the two scalar quantities in the following list:
force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton’s third law, relative velocity.

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