How to solve this question?

[lollipop]

Please help me to solve question 2

 

[lollipop]

The 1st question in the photo

 

[HallsofIvy]

Not only is it extremely hard to read (for one thing, the image is on its side) but part of the first line appears to be cut off.
I think the question is "if a single line passes through P, Q, R, and S, four points on the hyperbola x= a sec(t), y= b tan(t), then |PR|= |QS|". Is that correct?

 

[stapel]

I think the question is "if a single line passes through P, Q, R, and S, four points on the hyperbola x= a sec(t), y= b tan(t), then |PR|= |QS|". Is that correct?

You're close:

If the lines passing through $\displaystyle P$ and $\displaystyle Q$ on the hyperbola $\displaystyle \begin{cases}x\, =\, a\sec\left(\theta\right)\\y\, =\, b\tan\left(\theta\right)\end{cases}$ intersect with the asymptotes at $\displaystyle R$ and $\displaystyle S$ respectively, prove that $\displaystyle \left|PR\right|\, =\, \left|QS\right|$

You're quite right: That image is a bear to try to read! :shock:

 

[lollipop]

Yes! You guys are right! Anyone know how to solve? Teach me please

 

[HallsofIvy]

The first thing I would do is change to Cartesian coordinates: since $\displaystyle x= a sec(theta)$ and $\displaystyle y= b tan(\theta)$
$\displaystyle \frac{x^2}{a^2}- \frac{y^2}{b^2}= sec^2(\theta)- tan^2(\theta)= 1$. That is a hyperbola with center at (0, 0). For x and y very large compared with a and b, "1" is very small compared with the squares so, approximately, $\displaystyle \frac{x^2}{a^2}- \frac{y^2}{b^2}= 0$ which is the same as $\displaystyle \frac{x}{a}= \pm\frac{y}{b}$. Its asymptotes are $\displaystyle y= bx/a$ and $\displaystyle y= -bx/a$. Let P and Q be points on the hyperbola and write the equations of possible lines through them. Where do those cross the asymptotes?

 

[lollipop]

How to prove X3+X4=asec theta1 + asec theta2 ?

 

[stapel]

Okay; there's no way in heck I'm gonna try to read that sideways graphic! :shock: