Question

The mid point of the chord \( 4x - 3y = 5 \) of the hyperbola \( 2x^{2} - 3y^{2} = 12 \) is

• A. $(\frac{5}{4},0)$
• B. (2, 1)
• C. $(0,-\frac{5}{3})$
• D. $(\frac{11}{4},2)$

Hint:

Midpoint $(x_m, y_m)$ of a chord satisfies the chord equation. Check options to save time!

Answer 1

The Correct Option is B

Step 1: Intersection Points
Solve $y = \frac{4x-5}{3}$ and $2x^{2}-3y^{2}=12$. $2x^{2}-3(\frac{4x-5}{3})^{2}=12 \Rightarrow 2x^{2}-\frac{(4x-5)^{2}}{3}=12 \Rightarrow 6x^{2}-(16x^{2}-40x+25)=36$. $10x^{2}-40x+61=0$.
Step 2: Midpoint x-coordinate
The x-coordinate of the midpoint is the average of the roots: $x_{m} = \frac{x_{1}+x_{2}}{2} = \frac{-(-40)/10}{2} = \frac{4}{2} = 2$.
Step 3: Midpoint y-coordinate
Substitute $x=2$ into the chord equation $4x-3y=5$: $4(2)-3y=5 \Rightarrow 8-5=3y \Rightarrow 3y=3 \Rightarrow y=1$. The midpoint is $(2, 1)$.
Final Answer: (b)