Question

If the equations $3x^2+2hxy-3y^2=0$ and $3x^2+2hxy-3y^2+2x-4y+c=0$ represent the four sides of a square, then $\frac{h}{c}= $

• A. 1/4
• B. -2/3
• C. -3
• D. -4

Hint:

The equation $ax^2+2hxy+by^2=0$ represents perpendicular lines if $a+b=0$. If a general second degree equation has the same homogeneous part, it represents a pair of lines parallel to the first pair. For these four lines to form a square, additional conditions on the linear and constant terms must be met.

Answer 1

The Correct Option is D

Step 1: Analyze equations.
First represents perpendicular lines through origin; second represents parallel lines to first pair. Four lines form a square.

Step 2: Solve for $h$ and $c$.
Using line pair representation, distances between parallel lines equal, solve linear equations: \[ h=4, c=-1 \implies \frac{h}{c} = -4 \] \[ \boxed{-4} \]