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 (x1, y1) lies outside, on or inside the hyperbola x2a2 - y2b2 = 1 according as x21a2 - y21b2 – 1 < 0, = or > 0.

Let P (x1, y1) be any point on the plane of the hyperbola x2a2 - y2b2 = 1 ………………….. (i)

Position of a Point with Respect to the Hyperbola

From the point P (x1, y1) 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 (x1, y2).

Since the point Q (x1, y2) lies on the hyperbola x2a2 - y2b2 = 1.

Therefore,

x21a2 - y22b2 = 1        

y22b2 = x21a2 - 1 ………………….. (i)

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

PM <, = or > QM

i.e., according as y1 <, = or > y2

i.e., according as y21b2 <, = or > y22b2

i.e., according as y21b2 <, = or > x21a2 - 1, [Using (i)]

i.e., according as x21a2 - y21b2 <, = or > 1

i.e., according as x21a2 - y21b2 - 1 <, = or > 0

Therefore, the point

(i) P (x1, y1) lies outside the hyperbola x2a2 - y2b2 = 1 if PM < QM

i.e., x21a2 - y21b2 - 1 < 0.

(ii) P (x1, y1) lies on the hyperbola x2a2 - y2b2 = 1 if PM = QM

i.e., x21a2 - y21b2 - 1 = 0.

(ii) P (x1, y1) lies inside the hyperbola x2a2 - y2b2 = 1 if PM < QM

i.e., x21a2 - y21b2 - 1 > 0.

Hence, the point P(x1, y1) lies outside, on or inside the hyperbola x2a2 - y2b2 = 1 according as xx21a2 - y21b2  - 1 <, = or > 0.

Note:

Suppose E1 = x21a2 - y21b2 - 1, then the point P(x1, y1) lies outside, on or inside the hyperbola x2a2 - y2b2 = 1 according as E1 <, = or > 0.

Position of a Point with Respect to a Hyperbola

Solved examples to find the position of the point (x1, y1) with respect to an hyperbola x2a2 - y2b2 = 1:

1. Determine the position of the point (2, - 3) with respect to the hyperbola x29 - y225 = 1.

Solution:

We know that the point (x1, y1) lies outside, on or inside the hyperbola x2a2 - y2b2 = 1 according as

x21a2 - y21b2 – 1 < , = or > 0.

For the given problem we have,

x21a2 - y21b2 - 1 = 229 - (3)225 – 1 = 49 - 925 - 1 = - 206225 < 0.

Therefore, the point (2, - 3) lies outside the hyperbola x29 - y225 = 1.


2. Determine the position of the point (3, - 4) with respect to the hyperbola x29 - y216 = 1.  

Solution:

We know that the point (x1, y1) lies outside, on or inside the hyperbola x2a2 - y2b2 = 1 according as

x21a2 - y21b2 - 1 <, = or > 0.

For the given problem we have,

x21a2 - y21b2 - 1 = 329 - (4)216 - 1 = 99 - 1616 - 1 = 1 - 1 - 1 = -1 < 0.

Therefore, the point (3, - 4) lies outside the hyperbola x29 - y216 = 1.  

The Hyperbola






11 and 12 Grade Math 

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