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A line y=mx+1...

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 $P Q$ has $x$ -coordinate $- \frac{3}{5}$ , then which one of the following options is correct?

- 3 \leq m < - 1

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2 \leq m < 4

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4 \leq m < 6

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6 \leq m < 8

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Solution

The correct option is B $2 \leq m < 4$

Coordinate of midpoint
$R = (- \frac{3}{5}, - \frac{3}{5} m + 1) (S l o p e)_{P Q} \times (S l o p e)_{C R} = - 1 \Rightarrow m \times \frac{- \frac{3}{5} m + 1 + 2}{- \frac{3}{5} - 3} = - 1 \Rightarrow m \times \frac{m - 5}{6} = - 1 \Rightarrow m^{2} - 5 m + 6 = 0 \Rightarrow (m - 2) (m - 3) = 0 \Rightarrow m = 2, 3 ∴ 2 \leq m < 4$

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Standard XII Mathematics

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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 −35, then which one of the following options is correct?

The correct option is B 2≤m<4 Coordinate of midpoint R=(−35,−35m+1)(Slope)PQ×(Slope)CR=−1⇒m×−35m+1+2−35−3=−1⇒m×m−56=−1⇒m2−5m+6=0⇒(m−2)(m−3)=0⇒m=2,3∴2≤m<4

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 $P Q$ has $x$ -coordinate $- \frac{3}{5}$ , then which one of the following options is correct?