The chord of contact of tangents from a point `P` to a circle passes through `Qdot` If `l_1a n dl_2` are the length of the tangents from `Pa n dQ` to the circle, then `P Q` is equal to `(l_1+l_2)/2` (b) `(l_1-l_2)/2` `sqrt(l1 2+l2 2)` (d) `2sqrt(l1 2+l2 2)`

Options

A. `(l_(1)+l_(2))/(2)`
B. `(l_(2)-l_(2))/(2)`
C. `sqrt(l_(1)^(2) + l_(2)^(2))`
D. `sqrt(l_(1)^(2)l_(2)^(2))`

Answer

C

Explanation

Let `P(x_(1),y_(1)) and Q(x_(2), y_(2))` be two points and `x^(2)+y^(2)=a^(2)` be the given circle. Then, the chord of contact of tangents drawn from P to the given circle is `x x_(1)+y y_(1)=a^(2)`
It will pass through `Q(x_(2), y_(2))`, if
`x_(1)x_(2)+y_(1)y_(2)=a^(2)" " ...(i)`
Now, `l_(1)=sqrt(x_(1)^(2) + y_(1)^(2)-a^(2)), l_(2)=sqrt(x_(2)^(2)+y_(2)^(2)-a^(2))`
`:. PQ=sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2))`
`rArr PQ=sqrt((x_(2)^(2)+y_(2)^(2))+(x_(1)^(2)+y_(1)^(2))-2(x_(1)x_(2)+y_(1)y_(2)))`
`rArr PQ=sqrt((x_(2)^(2)+y_(2)^(2))+(x_(1)^(2)+y_(2)^(2))-2a^(2))`[Using (i)]
`rArr PQ=sqrt((x_(1)^(2)+y_(1)^(2)-a^(2))+(x_(2)^(2)+y_(2)^(2)-a^(2)))`
`rArr PQ=sqrt(l_(1)^(2)+l_(2)^(2))`