Question

two parallel tangents to a given circle are cut by a third tangents in the points R and Q show that the lines from R and Q to the centre of the circle are mutually perpendicular​

Answer 1

Step-by-step explanation:

Let the circle be x

2

+y

2

=a

2

and two parallel tangents be drawn at A(0,a) and at B(0,−a) whose equations are y=a and y=−a.

Let P(acosθ,asinθ) be any point on the circle tangent at which is xcosθ+ysinθ=a. Solving with the tangents y=a and y=−a, we get the points

Q(a

cosθ

1−sinθ

,a) and R(a

cosθ

1+sinθ

,−a)

and C is (0,0).

m

1

= slope of CQ=

a(1−sinθ)

acosθ

=

1−sinθ

cosθ

m

2

= slope of CR=

a(1+sinθ)

−acosθ

=

1+sinθ

−cosθ

Clearly m

1

m

2

=−1 as 1−sin

2

θ=cos

2

θ

Hence the lines CQ and CR are perpendicular.

solution

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