Question

If (5, 12) and (24, 7) are the foci of a conic passing through the origin, then the eccentricity of conic is
(a) $\dfrac{\sqrt{386}}{12}$
(b) $\dfrac{\sqrt{386}}{13}$
(c) $\dfrac{\sqrt{386}}{25}$
(d) $\dfrac{\sqrt{386}}{38}$

Answer 1

Hint: For solving this problem, first we have to calculate the distance between foci by using the distance formula. As we know that, for different conics we have different formulas for focal distance. By using various formulas, we can easily evaluate the eccentricity.

Complete step-by-step answer:
According to our question, we are given two coordinates for foci of a conic. Let us assume the coordinates of foci be S (5, 12) and S' (24, 7). Since the conics pass through the origin P(0, 0), therefore we calculate the focal distance by using the distance formula.
The distance between two coordinates $A\left( {{x}_{1}},{{y}_{1}} \right)\text{ and }B\left( {{x}_{2}},{{y}_{2}} \right)$ can be stated as: $AB=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$
Therefore, the distance between SS’ can be given as:
$\begin{align}
  & S{S}'=\sqrt{{{\left( 24-5 \right)}^{2}}+{{\left( 7-12 \right)}^{2}}} \\
 & S{S}'=\sqrt{{{19}^{2}}+{{(-5)}^{2}}} \\
 & S{S}'=\sqrt{361+25} \\
 & S{S}'=\sqrt{386} \\
\end{align}$
Also, the distance between SP can be given as:
$\begin{align}
  & SP=\sqrt{{{\left( 5-0 \right)}^{2}}+{{\left( 12-0 \right)}^{2}}} \\
 & SP=\sqrt{25+144} \\
 & SP=\sqrt{169} \\
 & SP=13 \\
\end{align}$
Distance between S’P can be given as:
$\begin{align}
  & {S}'P=\sqrt{{{\left( 24-0 \right)}^{2}}+{{\left( 7-0 \right)}^{2}}} \\
 & {S}'P=\sqrt{576+49} \\
 & {S}'P=\sqrt{625} \\
 & {S}'P=25 \\
\end{align}$
Also, the focal distance for any conic section is defined as 2ae.
$\therefore 2ae=\sqrt{386}\ldots (1)$
If the given conic section is ellipse then the sum of SP and S’P is equal to 2a.
For an ellipse:
 $\begin{align}
  & SP+{S}'P=25+13 \\
 & SP+{S}'P=38 \\
 & 2a=38\ldots (2) \\
\end{align}$
By using equation (1) and (2), we get
$\begin{align}
  & 38\times e=\sqrt{386} \\
 & e=\dfrac{\sqrt{386}}{38} \\
\end{align}$
If the given conic section is hyperbola then the difference of S’P and SP is equal to 2a.
For a hyperbola:
$\begin{align}
  & {S}'P-SP=25-13 \\
 & {S}'P-SP=12 \\
 & 2a=12\ldots (3) \\
\end{align}$
By using equation (1) and (3), we get
$\begin{align}
  & 12\times e=\sqrt{386} \\
 & e=\dfrac{\sqrt{386}}{12} \\
\end{align}$
Therefore, option (a) and option (d) are correct.

Note: The key concept involved in solving this problem is the knowledge of the formula of focal distance and the properties of hyperbola and ellipse. These concepts are very useful in solving complex problems. Their formula can be easily remembered and applied to a variety of questions.