Question

The locus of a point which is collinear with the point $(3,4)$ and $( - 4,3)$ is
A) $2x + 3y - 12 = 0$
B) $2x + 3y + 12 = 0$
C) $x + y + 12 = 0$
D) $x - 7y + 25 = 0$

Answer 1

Hint: If three points are collinear then they form a triangle with zero area. In this type of question we assume $P(x,y)$. Here we have to use the formula of slope. In this question points are collinear to each other so the slope of the first line with two coordinates is equal to the slope of the other line with two coordinates.

Complete step-by-step answer:
Here we have $P(x,y)$ is a point which is collinear with given point $A(3,4)$ and $B( - 4,3)$,
${x_1} = x$ , ${x_2} = 3$ , ${x_3} = - 4$
${y_1} = y$ , ${x_2} = 4$ , ${y_3} = 3$
We know the formula of slope of line
$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
So, we use the formula.
As the point is collinear and we have three points so the line made up with this point is two.
 Slope of $PA$ = Slope of $PB$
$\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$=$\dfrac{{{y_3} - {y_2}}}{{{x_3} - {x_2}}}$
$\dfrac{{4 - y}}{{3 - x}} = \dfrac{{3 - y}}{{ - 4 - x}}$
Use the method of cross multiplication.
$ - 16 - 4x + 4y + xy = 9 - 3x - 3y + xy$
Simplify the equation
$ - 4x + 3x + 4y + 3y = 9 + 16$
$7y - x = 25$
Evaluating the equation
$x - 7y + 25 = 0$
So, option D is the correct answer.

Note: If the three points are collinear then they form a triangle with a zero area.
Let $(x,y)$ be the coordinates of a point collinear with the given points.
Then, we use the formula of area of a triangle:
=$(\dfrac{1}{2})({x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2}))$
Here we put the values on the formula
=$\dfrac{1}{2}(3(3 - y) + ( - 4)(y - 4) + x(4 - 3)) = 0$
Because the RHS is 0 we can remove the $\dfrac{1}{2}$
After remove 0 we have equation
$3(3 - y) - 4(y - 4) + x(4 - 3) = 0$
Now we open the bracket
$9 - 3y - 4y + 16 + x = 0$
$25 = 7y - x$
We can also write it as $x - 7y + 25 = 0$