Question

Prove that the line 5x+12y=9 touches the hyperbola \[{{x}^{2}}-9{{y}^{2}}=9\] and find its point of contact.

Answer 1

Hint: To find that whether a curve or a line touches another line or curve, we follow the following procedure:-
1. We first make the coefficient of any one variable in any of the equations as 1.
2. Then we substitute this variable in the other curve’s or line’s equation.
3. This makes the entire equation in a single variable only.
4. Then we can solve for that variable and get the value of that variable.
5. Finally, we can put the value of the known variable in any of the equations and then get the point of contact as well because we have both the coordinates.

Complete Step-by-step answer:
As mentioned in the question, we have to find the point of contact of the given line and the given hyperbola and for that we would follow the procedure as mentioned in the hint which is as follows
\[\begin{align}
  & 5x+12y=9 \\
 & x=\dfrac{9-12y}{5} \\
\end{align}\]
 Now, we put this value in the equation of the hyperbola as follows

\[\begin{align}
  & {{x}^{2}}-9{{y}^{2}}=9 \\
 & \Rightarrow {{\left( \dfrac{9-12y}{5} \right)}^{2}}-9{{y}^{2}}=9 \\
 & \Rightarrow {{\left( 9-12y \right)}^{2}}-25\times 9{{y}^{2}}=9\times 25 \\
 & \Rightarrow 81+144{{y}^{2}}-216y-225{{y}^{2}}=225 \\
 & \Rightarrow -81{{y}^{2}}-216y-144=0 \\
 & \Rightarrow -9{{y}^{2}}-24y-16=0 \\
 & \Rightarrow 9{{y}^{2}}+24y+16=0 \\
\end{align}\]
On solving this quadratic equation, we get

\[\begin{align}
  & y=\dfrac{-24\pm \sqrt{{{24}^{2}}-4\cdot 9\cdot 16}}{2\cdot 9} \\
 & y=\dfrac{-24\pm \sqrt{576-576}}{18} \\
 & y=\dfrac{-24}{18} \\
 & y=-\dfrac{4}{3} \\
\end{align}\]
Now, putting this value in the equation of the line, we get
\[\begin{align}
  & x=\dfrac{9-12\times \dfrac{-4}{3}}{5} \\
 & x=\dfrac{25}{5} \\
 & x=5 \\
\end{align}\]
Hence, the line touches the hyperbola at
\[(x,y)=\left( 5,\dfrac{-4}{3} \right)\]

Note: The students can make an error if they don’t know the procedure to find the point of contact or intersection points as then they won’t be able to get to the correct value.
Another important thing is that if more than one point of contact appears, then this means that the two curves are intersecting and the points that we will get would be those points only which are the points of intersection.