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