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Question
Form the differential equation of all straight lines touching the circle x2 + y2 = r2
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Solution
Given circle equation be x2 + y2 = r2
Let y = mx + c be all straight lines which towards the given circle x2 + y2 = r2
The condition for y = mx + c ……. (1)
Be a tangent to the circle x2 + y2 = r2
Be c2 = r2(1 + m2)
⇒ c = `sqrt(1 + "m"^2)`
Substituting c value in equation (1), we get
y = `"mx" + "r" sqrt(1 + "m"^2)`
y – mx = `"r" sqrt(1 + "m"^2)` ......(2)
Differentiating equation (2) w.r.t x, we get
`("d"y)/("d"x) - "m"` = 0
`("d"y)/("d"x)` = m ........(3)
Substituting equation (3) in equation (2), we get
`y - x(("d"y)/("d"x)) = "r" sqrt(1 + (("d"y)/("d"x))^2`
Squaring on both sides, we get
`[y - x ("d"y)/"d"x]^2 = ["r" sqrt(1 + (("d"y)/("d"x))^2]]^2`
`[y - x ("d"y)/"d"x]^2 = "r"^2 [1 + (("d"y)/("d"x))^2]`
Which is a required differential equation.
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