Question

A straight line given by the equation  $\left|\begin{array}{ccc}x+3& y-1& 1\\ 7& -1& 1\\ -3& 9& 1\end{array}\right|=0$ passes through the point. 

• A. (4, -1)
• B. (-6, 0)
• C. (6, 10)
• D. (4, 0)

Answer 1

The correct option is D

(4, 0)

Given equation of straight line is,

$\left|\begin{array}{ccc}x+3& y-1& 1\\ 7& -1& 1\\ -3& 9& 1\end{array}\right|=0$

We know that the equation of a straight line passing through two points (x1,y1) and (x2,y2) is given by the equation,

$\left|\begin{array}{ccc}x& y& 1\\ x1& y1& 1\\ x2& y2& 1\end{array}\right|=0$

If we are able to convert the given determinant in the form above we will be able to directly obtain the two points (x1,y1) and (x2,y2). But these are just two of the infinite number of points lying on the line. 

Now, let’s do a couple of transformations on the original determinant to change it to the desired form,

$C_1\to C_1–3C_1{C}_2\to C_2+C_3$

$\left|\begin{array}{ccc}x& y& 1\\ 4& 0& 1\\ -6& 10& 1\end{array}\right|=0$

This is the equation of straight line passing through 2 points (4, 0) and (-6, 10). If options contains any of these point its straight forward. If not, expand the determinant to form the equation of line and see which option satisfies the equation. In this case (4,0) is given as an option. So the correct option is (d).