Question

Form the differential equation of all straight lines touching the circle x² + y² = r²

Answer 1

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.