Position of a Point with Respect to the Hyperbola

We will learn how to find the position of a point with respect to the hyperbola.

The point P (x$_{1}$, y$_{1}$) lies outside, on or inside the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 according as $\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ – 1 < 0, = or > 0.

Let P (x$_{1}$, y$_{1}$) be any point on the plane of the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 ………………….. (i)

From the point P (x$_{1}$, y$_{1}$) draw PM perpendicular to XX' (i.e., x-axis) and meet the hyperbola at Q.

According to the above graph we see that the point Q and P have the same abscissa. Therefore, the co-ordinates of Q are (x$_{1}$, y$_{2}$).

Since the point Q (x$_{1}$, y$_{2}$) lies on the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1.

Therefore,

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

$\frac{y_{2}^{2}}{b^{2}}$ = $\frac{x_{1}^{2}}{a^{2}}$ - 1 ………………….. (i)

Now, point P lies outside, on or inside the hyperbola according as

PM <, = or > QM

i.e., according as y$_{1}$ <, = or > y$_{2}$

i.e., according as $\frac{y_{1}^{2}}{b^{2}}$ <, = or > $\frac{y_{2}^{2}}{b^{2}}$

i.e., according as $\frac{y_{1}^{2}}{b^{2}}$ <, = or > $\frac{x_{1}^{2}}{a^{2}}$ - 1, [Using (i)]

i.e., according as $\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ <, = or > 1

i.e., according as $\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ - 1 <, = or > 0

Therefore, the point

(i) P (x$_{1}$, y$_{1}$) lies outside the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 if PM < QM

i.e., $\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ - 1 < 0.

(ii) P (x$_{1}$, y$_{1}$) lies on the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 if PM = QM

i.e., $\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ - 1 = 0.

(ii) P (x$_{1}$, y$_{1}$) lies inside the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 if PM < QM

i.e., $\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ - 1 > 0.

Hence, the point P(x$_{1}$, y$_{1}$) lies outside, on or inside the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 according as x$\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$  - 1 <, = or > 0.

Note:

Suppose E$_{1}$ = $\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ - 1, then the point P(x$_{1}$, y$_{1}$) lies outside, on or inside the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 according as E$_{1}$ <, = or > 0.

Solved examples to find the position of the point (x$_{1}$, y$_{1}$) with respect to an hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1:

1. Determine the position of the point (2, - 3) with respect to the hyperbola $\frac{x^{2}}{9}$ - $\frac{y^{2}}{25}$ = 1.

Solution:

We know that the point (x$_{1}$, y$_{1}$) lies outside, on or inside the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 according as

$\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ – 1 < , = or > 0.

For the given problem we have,

$\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ - 1 = $\frac{2^{2}}{9}$ - $\frac{(-3)^{2}}{25}$ – 1 = $\frac{4}{9}$ - $\frac{9}{25}$ - 1 = - $\frac{206}{225}$ < 0.

Therefore, the point (2, - 3) lies outside the hyperbola $\frac{x^{2}}{9}$ - $\frac{y^{2}}{25}$ = 1.

2. Determine the position of the point (3, - 4) with respect to the hyperbola $\frac{x^{2}}{9}$ - $\frac{y^{2}}{16}$ = 1.  

Solution:

We know that the point (x$_{1}$, y$_{1}$) lies outside, on or inside the hyperbola $\frac{x^{2}}{a^{2}}$ - $\frac{y^{2}}{b^{2}}$ = 1 according as

$\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ - 1 <, = or > 0.

For the given problem we have,

$\frac{x_{1}^{2}}{a^{2}}$ - $\frac{y_{1}^{2}}{b^{2}}$ - 1 = $\frac{3^{2}}{9}$ - $\frac{(-4)^{2}}{16}$ - 1 = $\frac{9}{9}$ - $\frac{16}{16}$ - 1 = 1 - 1 - 1 = -1 < 0.

Therefore, the point (3, - 4) lies outside the hyperbola $\frac{x^{2}}{9}$ - $\frac{y^{2}}{16}$ = 1.  

● The Hyperbola

• Definition of Hyperbola
• Standard Equation of an Hyperbola
  
• Vertex of the Hyperbola
• Centre of the Hyperbola
• Transverse and Conjugate Axis of the Hyperbola
• Two Foci and Two Directrices of the Hyperbola
• Latus Rectum of the Hyperbola
• Position of a Point with Respect to the Hyperbola
• Conjugate Hyperbola
• Rectangular Hyperbola
• Parametric Equation of the Hyperbola
• Hyperbola Formulae
• Problems on Hyperbola

11 and 12 Grade Math 

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