Question

5. If x² + y2 = c and x/a +y/b= 1 intersect at A and B then find AB. Hence deduce the
condition that the line touches the circle.​

Answer 1

so the chord intersects circle at A and B. if you drop a perpendicular from the centre O of circle on the chord at P, it will bisect AB (standard theorem).

hence AB= 2PA= 2*sqrt(r^2 – (OP)^2) by pyth theorem, where r is the radius of circle.

note that x²+y²=c² has centre O(0, 0) and radius = c.

so, AB= 2*sqrt(c^2 – (OP)^2).

distance of O(0, 0) from line X/a+y/b=1 or bx + ay – ab= 0 will be:

OP = |b*0 + a*0 – ab|/sqrt(b^2 + a^2)

so, AB= 2*sqrt(c^2 – a^2b^2/(b^2 + a^2)).

now, for tangency AB= 0 as tangent touches circle at one point only.

so, 2*sqrt(c^2 – a^2b^2/(b^2 + a^2))= 0

or c^2(b^2 + a^2) = a^2b^2

or 1/a^2 + 1/b^2 = 1/c^2

kindly approve :))

please mark me as brainliest answer.

and follow me also.