A line y = mx + 1 intersects the circle (x â 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?

Question

A line y = mx + 1 intersects the circle (x â 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? (A) -3 le m<-1 (B) 2 le m<4 (C) 4 le m<6 (D) 6 le m<8

Tardigrade Admin · Accepted Answer

The mid-point of chord PQ can be considered as A(-(3/5), -(3/5)m+1) ∵ AO ⊥ PQ Slope of AO Ã Slope of PQ = -1 ⇒ ((1-(3/5)m+2/-(3)5-3)).m=-1 ⇒ (3-(3/5)m)m=(18/5) ⇒ m2-5m+6=0 ⇒ m=2 or 3

Q. A line $y = m x + 1$ 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 $- \frac{3}{5}$ , then which one of the following options is correct?

$- 3 \leq m < - 1$

$2 \leq m < 4$

$4 \leq m < 6$

$6 \leq m < 8$

Q. 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 −53​, then which one of the following options is correct?

−3≤m<−1

2≤m<4

4≤m<6

6≤m<8

The mid-point of chord PQ can be considered as A(−53​,−53​m+1) ∵AO⊥PQ<br/>SlopeofAO×SlopeofPQ=−1 ⇒(−53​−31−53​m+2​).m=−1 ⇒(3−53​m)m=518​ ⇒m2−5m+6=0 ⇒m=2 or 3

Q. A line $y = m x + 1$ 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 $- \frac{3}{5}$ , then which one of the following options is correct?

$- 3 \leq m < - 1$

$2 \leq m < 4$

$4 \leq m < 6$

$6 \leq m < 8$