Question

Q.12. If the line 2x + y + k = 0 is tangent to the hyperbola
x
= 1, then find the value of k​

Answer 1

Answer:

2x-y+1=0

y=2x+1

equation of hyperbole

x^2/a^2 - y^2/b^2=1

c^2=a^2m^2-b^2

1^2=a^24-16

a^2=17/4

a=root of 17/4

(^2 is square of the numbers)

Answer 2

The value of k is ±2√2

Step-by-step explanation:

The equation of tangent line to hyperbola, 2x + y + k = 0

The hyperbola equation, xy = 1

Let point on hyperbola where tangent form (a,1/a)

$y=\dfrac{1}{x}$

$y'=-\dfrac{1}{x^2}$

Slope of tangent at x=a

$m=-\dfrac{1}{a^2}$

Equation of tangent to hyperbola,

$y-\dfrac{1}{a}=-\dfrac{1}{a^2}(x-a)$

$\dfrac{1}{a^2}x+y-\dfrac{2}{a}=0$

Given equation of tangent, 2x + y + k = 0

Compare both equation

So, $\dfrac{1}{a^2}=2$

$a=\pm\dfrac{1}{\sqrt{2}}$

$k=-\dfrac{2}{a}$

$k=\pm 2\sqrt{2}$

hence, the value of k is ±2√2

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