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JEE Advanced 2019 · Paper 1, Mathematics question 3

A line y=mx+1y = mx + 1 intersects the circle (x−3)2+(y+2)2=25(x-3)^2 + (y+2)^2 = 25 at the points PP and QQ. If the midpoint of the line segment PQPQ has xx-coordinate −35-\dfrac35, then which one of the following options is correct?

Choose one option
Answer and worked solution

Official answer: B

The line from the centre (3,−2)(3,-2) to the midpoint of the chord is perpendicular to the chord.

The midpoint. It lies on the line at x=−35x=-\frac35, so it is (−35, 1−3m5)\left(-\frac35,\,1-\frac{3m}5\right).

Perpendicularity.
1−3m5+2−35−3⋅m=−1  ⟹  m(15−3m)=18  ⟹  m2−5m+6=0.\frac{1-\frac{3m}5+2}{-\frac35-3}\cdot m=-1\implies m(15-3m)=18\implies m^2-5m+6=0.
So m=2m=2 or m=3m=3, both in [2,4)[2,4).

(B)

Topics:Equation of a circleAngle between lines
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