Question

the line y=x-10 intersects the curve x^2+y^2+4x+6y-40=0 at the point A and B. Find the length of the line AB. Please solve this I am stuck.

Answer 1

Answer:

7.071

Step-by-step explanation:

The line y=x-10 intersects the curve x²+y²+4x+6y-40=0 at points A and B. What is length of line AB?

The curve is a circle with centre  (−2,−3)  and radius  53−−√ . The point  (x,y)  on the line is distant  (x+2)2+(y+3)2−−−−−−−−−−−−−−−√=(x+2)2+(x−7)2−−−−−−−−−−−−−−−√=2x2−10x+53−−−−−−−−−−−−√=2(x−2.5)2+40.5−−−−−−−−−−−−−−√  from the centre. The minimum distance is therefore  40.5−−−−√ . So you have a chord  40.5−−−−√  from the centre of a circle with radius  53−−√ . The length of the chord is therefore of length  253−40.5−−−−−−−−√=50−−√=7.071 .