1.

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»

`6 le m LT 8`
`-3le m lt -1`
`4le m lt 6`
`2le m lt 4`

Solution :(Key idea : FIRSTLY find the centre of the given circle and write the coordinate of mid point (A),of line sequment PQ, since `Acbot PQ` THEREFORE use (slope of AC)x(slope of PQ)`=-1`.
It is given that points P and Q are intersecting points of circle.
`(x-3)^2+(y+2)^2=25` .....(i)
Line `"" y=mx+1 "".....(ii)`
And , the mid -point of PQ is A having x-coordinate `(-3)/(5)` so y-coordinate is `1-(3)/(5)m`.
So, `A(-(3)/(5),1-(3)/(5)M)`
From the figure,
`because AC BOT PQ`

`rArr` (slope of AC)x (slope of PQ)` =-1`.
`rArr ((-2-1+(3)/(5m))/(3+(3)/(5)))xxm=-I`
`rArr 3m^2-15m+18=0`
`rArr m=2or 3`.


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