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61. Find the equations of the tangent and normal to the hyperbola

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ at the point $(x_{0}, y_{0})$

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Vishal Baghel | Contributor-Level 10

4 months ago

The given eqⁿof the hyperbola is $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ______(1)

Differentiating eqn (1) wrt ‘x’ we get,

$\frac{2 x}{a^{2}} - \frac{2 y}{b^{2}} \frac{d y}{d x} = 0$

$\Rightarrow \frac{2 y}{b^{2}} \frac{d y}{d x} = \frac{2 x}{a^{2}}$

$\Rightarrow \frac{d y}{d x} = \frac{b^{2} x}{a^{2} y}$

${\frac{d y}{d x} |}_{(x, y)} = (x_{0}, y_{0}) \frac{b^{2} x_{0}}{a^{2} y_{0}}$ is the reqd slope of tangent to the curve

So, eqⁿof tangent at point $(x_{0}, y_{0})$ is

$y - y_{0} = \frac{b^{2} x_{0}}{a^{2} y_{0}} (x - x_{0}) .$

$\Rightarrow \frac{y y_{0}}{b^{2}} - \frac{{y_{0}}^{2}}{b^{2}} = \frac{x x_{0}}{a^{2}} - \frac{{x_{0}}^{2}}{a^{2}}$

$\Rightarrow \frac{x x_{0}}{a^{2}} - \frac{y y_{0}}{b^{2}} = \frac{{x_{0}}^{2}}{a^{2}} - \frac{{y_{0}}^{2}}{b^{2}}$

As $(x_{0}, y_{0})$ lies on the parabola given by eqn (1) we write,

$\frac{{x_{0}}^{2}}{a^{2}} - \frac{{y_{0}}^{2}}{b^{2}} = 1$

Hence, $\frac{x x_{0}}{a^{2}} - \frac{y y_{0}}{b^{2}} = 1$

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If the normal to the curve y(x) = ∫(from 2 to x) (2t² - 15t + 10)dt at a point (a, b) is parallel to the line x + 3y = -5, a > 1, then the value of |a + 6b| is equal to..........

Raj Pandey

y (x) = ∫? (2t² - 15t + 10)dt
dy/dx = 2x² - 15x + 10.
For tangent at (a, b), slope is m = dx/dy = 1 / (dy/dx) = 1 / (2a² - 15a + 10).
Given slope is -1/3.
2a² - 15a + 10 = -3
2a² - 15a + 13 = 0 (The provided solution has 2a²-15a+7=0, suggesting a different problem or a typo)
Following the image: 2a² - 15a + 7 = 0
(2a - 1) (a - 7) = 0
a = 1/2 or a = 7.
a = 1/2 Rejected as a > 1. So a = 7.
b = ∫? (2t² - 15t + 10)dt = [2t³/3 - 15t²/2 + 10t] from 0 to 7.
6b = [4t³ - 45t² + 60t] from 0 to 7 = 4 (7)³ - 45 (7)² + 60 (7) = 1372 - 2205 + 420 = -413.
|a + 6b| = |7 - 413| = |-406|

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If the tangent to the curve, y = f(x) = xlog? x, (x > 0) at a point (c, f(c)) is parallel to the line - segment joining the points (1,0) and (e, e) then c is equal to:

Raj Pandey

f' (c) = 1 + lnc = e/ (e-1)
lnc = e/ (e-1) - 1 = (e - (e-1)/ (e-1) = 1/ (e-1)
c = e^ (1/ (e-1)

The lengths of the sides of a triangle are 10 + x2, 10 + x2 and 20 – 2x2. If for x = k, the area of the triangle is maximum, then 3k2 is equal to:

Raj Pandey

$C D = \sqrt[]{{(10 + x^{2})}^{2} - {(10 - x^{2})}^{2}} = 2 \sqrt[]{10} | x |$

Area

$= \frac{1}{2} \times C D \times A B = \frac{1}{2} \times 2 \sqrt[]{10} | x | (20 - 2 x^{2})$

$\Rightarrow 10 - x^{2} = 2 x$

3x² = 10

x = k

3k² = 10

Let F₁(A,B,C) = $(A \land \sim B) \lor$ $[\sim C \land (A \lor B)] \lor \sim A a n d F_{2} (A, B) = (A \lor B) \lor (B \to \sim A)$ be two logical expressions. Then:

Vishal Baghel

By truth table

So F₁ (A, B, C) is not a tautology

Now again by truth table

So F₂ (A, B) be a tautology.

The triangle of maximum area that can be inscribed in a given circle of radius ‘r’ is:

Vishal Baghel

From option let it be isosceles where AB = AC then

$x = \sqrt[]{r^{2} - {(h - r)}^{2}}$

= $\sqrt[]{r^{2} - h^{2} - r^{2} + 2 r h}$

$x = \sqrt[]{2 h r - h^{2}} . . . . . . . . (i)$

Now ar $(Δ A B C) = Δ = \frac{1}{2} B C \times A L$

$Δ = \frac{1}{2} \times 2 \sqrt[]{2 h r - a^{2}} \times h$

then $x = \sqrt[]{2 \times \frac{3 r}{2} \times r - \frac{9 r^{2}}{4}} = \frac{\sqrt[]{3}}{2} r f r o m (i)$

$\Rightarrow B C = \sqrt[]{3} r$

So $A B = \sqrt[]{h^{2} + x^{2}} = \sqrt[]{\frac{9 r^{2}}{4} + \frac{3 r^{2}}{4}} = \sqrt[]{3} r$ .

Hence $Δ$ be equilateral having each side of length $\sqrt[]{3} r .$

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61. Find the equations of the tangent and normal to the hyperbola

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61. Find the equations of the tangent and normal to the hyperbola

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ at the point $(x_{0}, y_{0})$