Find the combined equation of the straight lines passing through the point $(1,1)$ and parallel to the lines represented by the equation $x^2-5xy+ 4y^2+\hspace{0.17em}x+ 2y-2=0$.

 

We are supposed to find the combined equation of a pair of straight lines through $(1,1)$ and parallel to another pair given by $x^2 - 5xy +\hspace{0.17em}4y^2 +\hspace{0.17em}x +2y-2=0$.

Since the required pair of lines is parallel to given pair of lines, their respective slopes must be equal.

For slope, we only consider the terms of second degree and equate them to zero, i.e., $x^2-5xy+4y^2 =0$.

$\Rightarrow (x-4y)(x-y) = 0$.

Hence, the two lines are parallel to the lines $x - 4y = 0, x-y =0$.

Thus, the two required lines are $x - 4y = c_1, x-y= c_2$.

Since, the two lines pass through $(1,1)$, on satisfying the coordinates of point in the above equations, we get,

$c_1 = \left(1\right)-4\left(1\right) = -3$ and $c_2 = \left(1\right) - \left(1\right) = 0$.

$\Rightarrow$Hence, the required lines are $x - 4y +\hspace{0.17em}3 =0 and x-y = 0$ and the combined equations is:

$(x-4y+3)(x-y) = x^2+4y^2-5xy + 3x - 3y = 0$ and option $1$ is correct.