Question

In a plane, \(\vec{a}\) and \(\vec{b}\) are the position vectors of two points \(A\) and \(B\) respectively. A point \(P\) with position vector \(\vec{r}\) moves on that plane in such a way that
\[ |\vec{r} - \vec{a}| - |\vec{r} - \vec{b}| = c \quad (\text{real constant}). \]
The locus of \(P\) is a conic section whose eccentricity is:

• A. \(\frac{|\vec{a} - \vec{b}|}{c}\)
• B. \(\frac{|\vec{a} + \vec{b}|}{c}\)
• C. \(\frac{|\vec{a} - \vec{b}|}{2c}\)
• D. \(\frac{|\vec{a} + \vec{b}|}{2c}\)

Hint:

For conic sections, remember the relationship between eccentricity and focal distances for ellipses, hyperbolas, and parabolas.

Answer 1

The Correct Option is A

1. The equation \(|\vec{r} - \vec{a}| - |\vec{r} - \vec{b}| = c\) describes the path of a point where the difference in its distances from fixed points \(A\) and \(B\) remains constant.

2. This defines a hyperbola, characterized by:

\(e = \frac{\text{Distance between foci}}{\text{Length of the transverse axis}}\)

3. The foci are separated by a distance of \(|\vec{a} - \vec{b}|\), and the transverse axis has a length of \(2c\).

4. Consequently, the eccentricity \(e\) is:

\(e = \frac{|\vec{a} - \vec{b}|}{c}\)