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  • P and Q are two points on a line passing through (2,4) and having slope m. If a line segment AB subtends a right angle at P and Q, where A(0,0) and B(6,0), then range of values of m is<div clas

P and Q are two points on a line passing through (2,4) and having slope m. If a line segment AB subtends a right angle at P and Q, where A(0,0) and B(6,0), then range of values of m is

Option: 1

\left(\frac{2-3 \sqrt{2}}{4}, \frac{2+3 \sqrt{2}}{4}\right)


Option: 2

\left(-\infty, \frac{2-3 \sqrt{2}}{4}\right) \cup\left(\frac{2+3 \sqrt{2}}{4}, \infty\right)


Option: 3

(-4,4)


Option: 4

(-\infty,-4) \cup(4, \infty)


Answers (1)

best_answer

A B subtends right angle at P and Q on variable line. So, AB is a diameter of circle
whose chord is a variable line. Equation of circle is x(x – 6) + y × y = 0

\mathrm{\Rightarrow x^2+y^2-6 x=0}              ....[i]

Equation of line through (2, 4) is y – 4 = m(x – 2)
⇒ y = mx + (4 – 2m) ...(ii)

\mathrm{\text { Line (ii) is a chord if }\left|\frac{3 m+(4-2 m)}{\sqrt{1+m^2}}\right|<3}

\mathrm{\begin{aligned} & \Rightarrow\left|\frac{4+m}{\sqrt{1+m^2}}\right|<3 \Rightarrow 16+8 m+m^2<9+9 m^2 \\ & \Rightarrow 8 m^2-8 m-7>0 \\ & \Rightarrow m \in\left(-\infty, \frac{2-3 \sqrt{2}}{4}\right) \cup\left(\frac{2+3 \sqrt{2}}{4}, \infty\right) \end{aligned}}

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SANGALDEEP SINGH

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