If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P…

Question

If a tangent to the circle x2 + y2  = 1 intersects
the coordinate axes at distinct points P and Q,
then the locus of the mid-point of PQ is :

Accepted Answer

Let the point of tangency A(cos $$\theta $$, sin $$\theta $$)

Equation of tangent at A,

xcos $$\theta $$ + ysin $$\theta $$ = 1

$$ \therefore $$ P(sec$$\theta $$, 0) and Q(0, cosec$$\theta $$)

Let M(h, k) is the mid-point of PQ.

$$ \therefore $$ h = $${{\sec\theta  + 0} \over 2}$$

$$ \Rightarrow $$ 2h = sec $$\theta $$

$$ \Rightarrow $$ cos $$\theta $$ = $${1 \over {2h}}$$ .....(1)

$$ \therefore $$ k = $${{\cos ec\theta  + 0} \over 2}$$

$$ \Rightarrow $$ 2k = cosec $$\theta $$

$$ \Rightarrow $$ sin $$\theta $$ = $${1 \over {2k}}$$ .....(2)

From (1) and (2),

sin2 $$\theta $$ + cos2 $$\theta $$ = $${\left( {{1 \over {2h}}} \right)^2} + {\left( {{1 \over {2k}}} \right)^2}$$

$$ \Rightarrow $$ $${1 \over {4{h^2}}} + {1 \over {4{k^2}}} = 1$$

$$ \Rightarrow $$ $${{{h^2} + {k^2}} \over {4{h^2}{k^2}}} = 1$$

$$ \Rightarrow $$ ${h^2} + {k^2} = 4{h^2}{k^2}$

$$ \therefore $$ Locus of the midpoint,

x2 + y2 -- 4x2y2 = 0

Solutions · 1