Given hyperbola $\left(\frac{x^2}{9}\right) - \left(\frac{y^2}{4}\right) = 1$ , find equation of line that passes through $M(5,1)$ and this point is the middle of line. I have no idea where to start the problem.
Given a hyperbola , find equation of chord that passes through given point
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discrete-mathematics
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0I don't understand the question, point $M$ is not on the hyperbola? – 2017-02-21
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0point **M** is the middle of the chord.So we need to find equation of this chord that passes through point **M** and this point is the middle of chord – 2017-02-21
1 Answers
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Hint: A property of hyperbolas that might be useful for solving this problem is that if $P$ and $Q$ are points on the hyperbola, and the line through these points intersects the asymptotes at $A$ and $B$, then the segments $\overline{AB}$ and $\overline{PQ}$ have the same midpoint. This allows you to change the problem from one of finding distance to intersections of a line with a hyperbola to an algebraically simpler one that involves distances to intersections with a pair of straight lines.
One way to proceed, then, is to take a variable point $P(t)$ on one of the asymptotes and compute the intersection $Q(t)$ of the line $\overline{PM}$ with the other asymptote. Compute the midpoint $(P+Q)/2$ and set this equal to $M$. This will give you a pair of quadratic equations in $t$, which, if they have a solution, will give you three points (including $M$) from which you can build the required line equation.
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0Thank you , this is very useful! – 2017-02-22