Question

O is the origin and A is the point $(3,4)$ .If a point P moves so that the line segment OP is always parallel to the line segment OA, then find the equation to the locus of P.
A.$4x - 3y = 0$
B. $4x + 3y = 0$
C. $3x + 4y = 0$
D. $3x - 4y = 0$

Answer 1

Hint: Draw a diagram of the stated problem and conclude that O, A and P are collinear. Then use the formula of collinearity to obtain the locus of P.

Formula Used:
If three points $({x_1},{y_1}),({x_2},{y_2}),({x_3},{y_3})$ are collinear then they satisfy the condition
${x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2}) = 0$

Complete step by step solution:
Suppose the coordinate of the point P is $(h,k)$ .
The diagram of the stated problem is,

Image: Line POA
That is, the given condition is only possible if the points O, A and P are collinear.
So, the points $O(0,0),A(3,4),P(h,k)$satisfies the condition
${x_1}({y_2} - {y_3}) + {x_2}({y_3} - {y_1}) + {x_3}({y_1} - {y_2}) = 0$.
Hence,
$0(4 - k) + 3(k - 0) + h(0 - 4) = 0$
$3k - 4h = 0$
$4h - 3k = 0$
So, the required locus is $4x - 3y = 0$.

Option ‘A’ is correct

Additional information:
Locus of a point is a set of points that satisfy an equation of curve. The curve may be a circle or hyperbola or ellipse or a line etc. The term "location" is derived from the locus. Locus denotes the location of something. The locus defines when an object is placed somewhere or something at a location. For example, the area has become a locus of resistance to the point.

Note: If three points lie on a line, then the area made by three points becomes zero. Since the points lie on a line, the area made by the points will be zero. At the end, replace h by x and k by y.