With usual notations the P(r, θ ) be representing the coordinates of a point in polar system. The equation of a line in polar form which is parallel to the line 
= a cos θ + b sin θ is of the form : 
= a cos 
+ b sin 
The distance of P(r 1 , θ 1 ) from 
= a cos θ + b sin θ is 
.Distance between 2 points (r 1 , θ 1 ), (r 2 , θ 2 ) is 
Equation of the tangent to the circle r = 2a cos θ at T( α ) on the circle is r cos ( θ – 2 α ) = 2a cos 2 α . General equation of the conic is of the form 
= 1 + e cos θ or 
= 1– e cos θ , e being the eccentricity of the conic, λ is the length of the semi latus-rectum.
(i) A possible value of θ so that the area of the triangle formed by (2, θ ), 
and 
is 
is –
Text Solution
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Ans.
(i)
Sol. With θ = π /2, the given points form a triangle with area 5 
/4 sq. units.
(ii)
Sol. We observe that the area of the triangle (with the formula given in passage) is '0'. Hence the points are collinear.
(iii)
Sol. r 2 sin 2 θ = c 2 ⇒ (r cos θ ) (r sin θ ) = c 2 /2 ⇒ xy = c 2 /2 is representing a rectangular hyperbola whose eccentricity is 
.
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