How many such points exists that are at a distance k from point P, which are also on the locus of the line ax + by + c = 0.
How many such points exists that are at a distance k from point P, which are also on the locus of the line ax + by + c = 0.
The correct option is B 0, 1 or 2
For this question we will divide the answer based on 3 conditions. The collection of all points at a distance k from point P will form a circle of radius k.
Condition 1: When k is lesser than the perpendicular distance from the line to the point. We can say that the radius of the circle is less than the perpendicular distance from the line to the centre of the circle i.e. point P. Here k < m, m is the perpendicular distance from the line to the point P. You can observe that the circle does not intercept the line ax + by + c = 0. So in this case, there are no points which are at a distance k from point P and also on line ax + by + c = 0.
Condition 2: When k is equal to the perpendicular distance from the line ax + by + c = 0, i.e. k = m. If we visualize the locus of the points at distance k from the point P as a circle, we get the following figure, If k = m, it means that the line is at a distance equal to the radius of the circle or that the line is just touching the circle. So there will be only one point which is at a distance k from the point P and lies on the line, if the perpendicular distance of the point from the line is equal to k.
Condition 3: If the distance k is larger than the perpendicular distance m from the line ax + by + c = 0, i.e. k > m. You can assume the locus of the points at a distance k from point P to be a circle as in the figure. When k > m, it means that the radius of the circle is larger than the perpendicular distance to the line to the point P, then the line becomes the secant to the circle. Thus intersect the circle at two distinct points. So we can conclude that there can be zero, one or two points at a distance k from any point P which lies on a line ax + by + c = 0, depending on the perpendicular distance of the line and the point P.
0, 1 or 2
For this question we will divide the answer based on 3 conditions. The collection of all points at a distance k from point P will form a circle of radius k.
Condition 1: When k is lesser than the perpendicular distance from the line to the point. We can say that the radius of the circle is less than the perpendicular distance from the line to the centre of the circle i.e. point P. Here k < m, m is the perpendicular distance from the line to the point P. You can observe that the circle does not intercept the line ax + by + c = 0. So in this case, there are no points which are at a distance k from point P and also on line ax + by + c = 0.
Condition 2: When k is equal to the perpendicular distance from the line ax + by + c = 0, i.e. k = m. If we visualize the locus of the points at distance k from the point P as a circle, we get the following figure, If k = m, it means that the line is at a distance equal to the radius of the circle or that the line is just touching the circle. So there will be only one point which is at a distance k from the point P and lies on the line, if the perpendicular distance of the point from the line is equal to k.
Condition 3: If the distance k is larger than the perpendicular distance m from the line ax + by + c = 0, i.e. k > m. You can assume the locus of the points at a distance k from point P to be a circle as in the figure. When k > m, it means that the radius of the circle is larger than the perpendicular distance to the line to the point P, then the line becomes the secant to the circle. Thus intersect the circle at two distinct points. So we can conclude that there can be zero, one or two points at a distance k from any point P which lies on a line ax + by + c = 0, depending on the perpendicular distance of the line and the point P.
