Question

If the curve ay + x² = 7 and x³ = y, cut orthogonally at (1, 1), then the value of a is ______.

विकल्प

• 1

• 0

• – 6

• 6

Answer 1

If the curve ay + x² = 7 and x³ = y, cut orthogonally at (1, 1), then the value of a is 6.

Explanation:

Equation of the given curves are ay + x² = 7  .....(i)

And x³ = y  .....(ii)

Differentiating eq. (i) w.r.t. x, we have

`"a" "dy"/"dx" + 2x` = 0

⇒ `"dy"/"dx" = - (2x)/"a"`

∴ m₁ = `- (2x)/"a"` ......`("m"_1 = "dy"/"dx")`

Now differentiating eq. (ii) w.r.t. x, we get

3x² = `"dy"/"dx"`

⇒ m₂ = `3x^2`  .....`("m"_2 = "dy"/"dx")`

The two curves are said to be orthogonal if the angle between the tangents at the point of intersection is 90°.

∴ m₁ × m₂ = – 1

⇒ `(-2x)/"a" xx 3x^2` = – 1

⇒ `(-6x^3)/"a"` = – 1

⇒ 6x³ = a

(1, 1) is the point of intersection of two curves.

∴ 6(1)³ = a

So a = 6