If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$ , then the locus of the mid point of $AB$ is :
- $x^2 - 4y^2 + 16x^2y^2 = 0$
- $x^2 - 4y^2 - 16x^2y^2 = 0$
- $4x^2 - y^2 + 16x^2y^2 = 0 $
- $4x^2 - y^2 - 16x^2y^2 = 0$
The Correct Option is B
Solution and Explanation
Point $ 4 y y_{1}=x x_{1}+1$ with $4 y_{1}^{2}=x_{1}^{2}+1$
x axis $\frac{-1}{x}, 0] $
y axis $\left[0, \frac{1}{4 y_{1}}\right] $
Mid point $h=\frac{-1}{2 x}, k=\frac{1}{8 y_{1}}$
$x_{1}=\frac{-1}{2 h} y_{1}=\frac{1}{8 k} $
$4\left(\frac{1}{8 k}\right)^{2}=\left(\frac{-1}{2 h}\right)^{2}+1 $
$\frac{4}{4 k^{2}}=\frac{1}{4 b^{5}}+1 $
$\frac{1}{16 y^{2}}=\frac{1}{4 b^{2}}+1 $
$\frac{1}{16 y^{2}}=\frac{1+4 x^{2}}{4 x^{2}}$
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
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