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 :

$4 y^{2}=x^{2}+1 $ 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 $ $\frac{1}{16 y^{2}}=\frac{1+4 x^{2}}{4 x^{2}}$

If the tangents drawn to the hyperbola $4y^2 = x^2

Questions Asked in JEE Main exam

View More Questions

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.

Read More: Application of Derivatives