Question

Find the condition that the curves 2x = y² and 2xy = k intersect orthogonally.

Answer 1

The two circles intersect orthogonally if the angle between the tangents drawn to the two circles at the point of their intersection is 90°.

Equation of the two circles are given as

2x = y²    ......(i)

And 2xy = k   ......(ii)

Differentiating eq. (i) and (ii) w.r.t. x, we get

2.1 = `2y * "dy"/"dx"`

⇒ `"dy"/"dx" = 1/y`

⇒ m₁ = `1/y`   ......(m₁ = slope of the tangent)

⇒ 2xy = k

⇒ `2[x * "dy"/"dx" + y * 1]` = 0

∴ `"dy"/"dx" = - y/x`

⇒ m₂ = `- y/x` ......[m₂ = slope of the other tangent]

If the two tangents are perpendicular to each other,

Then m₁ × m₂ = – 1

⇒ `1/y xx (- y/x)` = – 1

⇒ `1/x` = 1

⇒ x = 1

Now solving 2x = y²  ......[From (i)]

And 2xy = k   .....[From (ii)]

From equation (ii)

y = `"k"/(2x)`

Putting the value of y in equation (i)

2x = `("k"/(2x))^2`

⇒ 2x = `"k"^2/(4x^2)`

⇒ 8x³ = k²

⇒ 8(1)³ = k²

⇒ 8 = k²

Hence, the required condition is k² = 8.