Question

Paragraph :
Let $S$ and $S^{\prime }$ be the foci on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $P(x_1,y_1)$ be an arbitrary point as shown.
(a) Find the gradients of the line $PS$ and $P{S}^{\prime }$.

• A. $\frac{y_1}{x_1-ae}$ and $\frac{y_1}{x_1+ae}$ respectively.
• B. $\frac{y_1}{x_1+ae}$ and $\frac{y_1}{x_1-ae}$ respectively.
• C. $\frac{y_1+ae}{x_1}$ and $\frac{y_1-ae}{x_1}$ respectively.
• D. $\frac{y_1-ae}{x_1}$ and $\frac{y_1+ae}{x_1}$ respectively.

Answer 1

The correct option is A $\frac{y_1}{x_1-ae}$ and $\frac{y_1}{x_1+ae}$ respectively.

$P(x_1,y_1),S(ae,0),S^{\prime }(-ae,0)$
Clearly from the diagram,
Gradient of $PS=\frac{y_1}{x_1-ae}$
Gradient of $P{S}^{\prime }=\frac{y_1}{x_1+ae}$
Hence, correct option is A.