A line intersects the circle at the points P and Q. If the midpoint of the line segment PQ has -coordinate , then which one of the following options is correct? [2019]

Question

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? [2019]

Smart Achievers · Accepted Answer

(1)

Given: Circle ${(x - 3)}^{2} + {(y + 2)}^{2} = 25$ , with centre $C (3, - 2)$ and radius 5 is intersected by a line $y = m x + 1$ at P & Q such that coordinates of midpoint R of PQ is $- \frac{3}{5}$ .

Since $x$ -coordinate of point R is $- \frac{3}{5}$ and point R lies on the line $y = m x + 1$ , therefore $y$ -coordinate of R will be $\frac{3 m}{5} + 1$ .

$∴$ $R (- \frac{3}{5}, - \frac{3 m}{5} + 1)$

Since R is the midpoint of PQ, therefore $C R ⊥ P Q$

$\Rightarrow \frac{- \frac{3 m}{5} + 1 + 2}{- \frac{3}{5} - 3} \times m = - 1$

$\Rightarrow m^{2} - 5 m + 6 = 0 \Rightarrow m = 2, 3$

Solution

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 [2019]

1 $2 \leq m < 4$

2 $- 3 \leq m < - 1$

3 $4 \leq m < 6$

4 $6 \leq m < 8$

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Solution

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 -35, then which one of the following options is correct [2019]

1 2≤m<4

2 -3≤m<-1

3 4≤m<6