Question

if the point P has coordinates cos alpha sin alpha find the distance of op where O is the origin​

Answer 1

Answer:

Step-by-step explanation:

Key points to solve the problem:

• Idea of distance formula- Distance between two points P(x1,y1) and Q(x2,y2) is given by- PQ =

Given,

Two points P and Q subtends an angle α at the origin as shown in figure:

From figure we can see that points O,P and Q forms a triangle.

Clearly in ΔOPQ we have:

{from cosine formula in a triangle}

⇒

…..equation 1

From distance formula we have-

OP =

As, coordinates of O are (0, 0) ⇒ x2 = 0 and y2 = 0

Coordinates of P are (x1, y1) ⇒ x1 = x1 and y1 = y1

=

=

Similarly, OQ =

=

And, PQ =

∴ OP2 + OQ2 - PQ2 =

⇒ OP2 + OQ2 - PQ2 =

Using (a-b)2 = a2 + b2 – 2ab

∴ OP2 + OQ2 - PQ2 = 2x1 x2 + 2y1 y2 ….equation 2

From equation 1 and 2 we have:

⇒

…Proved.