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A line y=mx +1 intersects the circle (x-3)^2+(y+2)^2=25 at the points P and Q . If the midpoint of the line segment PQ has x-coordinate -(3)/(5), then which one of the following options is correct ? |
Answer» <html><body><p>`6 le m <a href="https://interviewquestions.tuteehub.com/tag/lt-537906" style="font-weight:bold;" target="_blank" title="Click to know more about LT">LT</a> 8` <br/>`-3le m lt -1` <br/>`4le m lt 6` <br/>`2le m lt 4` </p>Solution :(Key idea : <a href="https://interviewquestions.tuteehub.com/tag/firstly-462152" style="font-weight:bold;" target="_blank" title="Click to know more about FIRSTLY">FIRSTLY</a> find the centre of the given circle and write the coordinate of mid point (A),of line sequment PQ, since `Acbot PQ` <a href="https://interviewquestions.tuteehub.com/tag/therefore-706901" style="font-weight:bold;" target="_blank" title="Click to know more about THEREFORE">THEREFORE</a> use (slope of AC)x(slope of PQ)`=-1`. <br/> It is given that points P and Q are intersecting points of circle. <br/> `(x-3)^2+(y+2)^2=25` .....(i) <br/> Line `"" y=mx+1 "".....(ii)` <br/> And , the mid -point of PQ is A having x-coordinate `(-3)/(<a href="https://interviewquestions.tuteehub.com/tag/5-319454" style="font-weight:bold;" target="_blank" title="Click to know more about 5">5</a>)` so y-coordinate is `1-(3)/(5)m`. <br/> So, `A(-(3)/(5),1-(3)/(5)M)`<br/> From the figure, <br/> `because AC <a href="https://interviewquestions.tuteehub.com/tag/bot-25566" style="font-weight:bold;" target="_blank" title="Click to know more about BOT">BOT</a> PQ` <br/> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/41Y_SP_MATH_SP_2019_E01_003_S01.png" width="80%"/> <br/> `rArr` (slope of AC)x (slope of PQ)` =-1`.<br/> `rArr ((-2-1+(3)/(5m))/(3+(3)/(5)))xxm=-I` <br/> `rArr 3m^2-15m+18=0` <br/> `rArr m=2or 3`.</body></html> | |