230. Show that the tangent at the point \( (1,1) \) on the rectangular

1.	230. Show that the tangent at the point \( (1,1) \) on the rectangular hyperbola \( x y=1 \) cuts equal length of the axes.[Eco. (H)1996, Eco. (H) I Sem. 2014]
Answer» xy = 1 ⇒ y = 1/x ⇒ dy/dx = -1/x² ⇒ (dy/dx)_{(a + x = 1, y = 1)} = -1/1 = -1 \(\therefore\) slope of tangent of hyperbola xy = 1 at point (1, 1) = m = -1 \(\therefore\) Equation of tangent is y - y₁ = m(x - x₁) ⇒ y - 1 = -1(x - 1) (\(\because\) Tangent is find at point (1, 1)) ⇒ y - 1 = 1 - x ⇒ x + y = 2 ⇒ x/2 + y/2 = 1 \(\therefore\) a = x - intercept = 2 b = y-intercept = 2 \(\because\) a = b Hence, the tangent at the point (1, 1) on the rectangular hyperbola xy = 1 cuts equal length of the axes. Hence Proved.

230. Show that the tangent at the point \( (1,1) \) on the rectangular hyperbola \( x y=1 \) cuts equal length of the axes.[Eco. (H)1996, Eco. (H) I Sem. 2014]

Answer»

xy = 1

⇒ y = 1/x

⇒ dy/dx = -1/x²

⇒ (dy/dx)_{(a + x = 1, y = 1)} = -1/1 = -1

\(\therefore\) slope of tangent of hyperbola xy = 1 at point (1, 1) = m = -1

\(\therefore\) Equation of tangent is

y - y₁ = m(x - x₁)

⇒ y - 1 = -1(x - 1)

(\(\because\) Tangent is find at point (1, 1))

⇒ y - 1 = 1 - x

⇒ x + y = 2

⇒ x/2 + y/2 = 1

\(\therefore\) a = x - intercept = 2

b = y-intercept = 2

\(\because\) a = b

Hence, the tangent at the point (1, 1) on the rectangular hyperbola xy = 1 cuts equal length of the axes.

Hence Proved.

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