I have a collection of some “random”, yet what I call ‘beautiful” questions in Co-ordinate Geometry. I hope kids preparing for IITJEE Mains or KVPY or ISI Entrance Examination will also like them.

Problem 1:

Given n straight lines and a fixed point O, a straight line is drawn through O meeting lines in the points , , , , and on it a point R is taken such that

Show that the locus of R is a straight line.

Solution 1:

Let equations of the given lines be , , and the point O be the origin .

Then, the equation of the line through O can be written as where is the angle made by the line with the positive direction of x-axis and r is the distance of any point on the line from the origin O.

Let be the distances of the points from O which in turn and , where .

Then, coordinates of R are and of are where .

Since lies on , we can say for

, for

…as given…

Hence, the locus of R is which is a straight line.

Problem 2:

Determine all values of for which the point lies inside the triangle formed by the lines , , .

Solution 2:

Solving equations of the lines two at a time, we get the vertices of the given triangle as: , and .

So, AB is the line , AC is the line and BC is the line

Let be a point inside the triangle ABC. (please do draw it on a sheet of paper, if u want to understand this solution further.) Since A and P lie on the same side of the line , both and must have the same sign.

or which in turn which in turn either or ….call this relation I.

Again, since B and P lie on the same side of the line , and have the same sign.

and , that is, …call this relation II.

Lastly, since C and P lie on the same side of the line , we have and have the same sign.

that is

or ….call this relation III.

Now, relations I, II and III hold simultaneously if or .

Problem 3:

A variable straight line of slope 4 intersects the hyperbola at two points. Find the locus of the point which divides the line segment between these two points in the ratio .

Solution 3:

Let equation of the line be where c is a parameter. It intersects the hyperbola at two points, for which , that is, .

Let and be the roots of the equation. Then, and . If A and B are the points of intersection of the line and the hyperbola, then the coordinates of A are and that of B are .

Let be the point which divides AB in the ratio , then and , that is, …call this equation I.

and ….call this equation II.

Adding I and II, we get , that is,

….call this equation III.

Subtracting II from I, we get

so that the locus of is

More later,

Nalin Pithwa.