Question

If $2x+3y=7$ makes equal angle with $9x^2+12xy+k{y}^2=0$ then $k=$

• A. $3$
• B. $7$
• C. $14$
• D. $0$

Answer 1

The correct option is C $14$

Given eqs are

$2x+3y=7$...(1)

and $9x^2+12xy+k{y}^2=0$

Since eq (1) & eq(2) makes equal angle

$\therefore$ eq (1) is parallel to angle bisectors of eq(2)

from eq (1) $2x+3y=7$

$y=\frac{-2}{3}x+\frac{7}{3}$

Slope $m=-\frac{2}{3}$

from eq (2) $9x^2+12xy+k{y}^2=0$

$\Rightarrow \frac{x^2-y^2}{9-k}=\frac{xy}{6}$

$\Rightarrow \frac{1-(y/x{)}^2}{9-k}=\frac{y/x}{6}$

$\Rightarrow \frac{1-m^2}{9-k}=\frac{m}{6}$

$\Rightarrow \frac{1-4/9}{9-k}=\frac{-2/3}{6}$

$\Rightarrow \frac{5}{9(9-k)}=\frac{-2}{18}$

$\Rightarrow 5=-(9-k)$

$\Rightarrow k=14$