Question

Let \(T\) be the line passing through the points \(P(-2,7)\) and \(Q(2,-5)\). Let \(F_{1}\) be the set of all pairs of circles \(\left(S_{1}, S_{2}\right)\) such that \(T\) is tangent to \(S_{1}\) at \(P\) and tangent to \(S_{2}\) at \(Q\), and also such that \(S_{1}\) and \(S_{2}\) touch each other at a point, say, \(M .\) Let \(E_{1}\) be the set representing the locus of \(M\) as the pair \(\left(S_{1}, S_{2}\right)\) varies in \(F_{1}\). Let the set of all straight line segments joining a pair of distinct points of \(E_{1}\) and passing through the point \(R(1,1)\) be \(F_{2}\). Let \(E_{2}\) be the set of the mid-points of the line segments in the set \(F_{2}\). Then, which of the following statement(s) is (are) TRUE? (A) The point \((-2,7)\) lies in \(E_{1}\) (B) The point \(\left(\frac{4}{5}, \frac{7}{5}\right)\) does NOT lie in \(E_{2}\) (C) The point \(\left(\frac{1}{2}, 1\right)\) lies in \(E_{2}\) (D) The point \(\left(0, \frac{3}{2}\right)\) does NOT lie in \(E_{1}\)

Short answer

(A) is TRUE because P(-2,7) is the point where S1 is tangent. (B) is FALSE as (4/5, 7/5) lies on the line parallel to E1 through R. (C) is TRUE because (1/2, 1) lies on that same line through R. (D) is TRUE, (0, 3/2) does not satisfy the equation of E1.

Step-by-step solution

- Calculate the slope and equation of the line T

To start solving the problem, first determine the slope of line T using the coordinates of points P and Q. The slope, m, is given by the formula m = (y2 - y1) / (x2 - x1). Then use one of the points and the slope to write the equation of line T in slope-intercept form, y = mx + b.

- Identify the characteristics of circles S1 and S2

Since T is tangent to both circles at points P and Q, the radius of circle S1 will be perpendicular to T at P, and similarly for circle S2 at Q. The radii will be the lines extending from P and Q to the center of each circle, respectively. Since S1 and S2 touch each other at point M, M is on the line that passes through the centers of both S1 and S2.

- Determine the set E1

To find the locus of point M, notice that as S1 and S2 vary but remain touching and tangent to line T at P and Q, M will trace out the perpendicular bisector of segment PQ. This is because M is always equidistant from P and Q, by the definition of a tangent. The perpendicular bisector of PQ can be found using the midpoint of PQ and the negative reciprocal of the slope of T.

- Describe the set F2

The set F2 consists of all the line segments joining distinct points of E1 that pass through point R. These line segments form a continuous set of lines that pass through R and are symmetric about the perpendicular bisector of PQ (which is the set E1).

- Find the set E2

The set E2 is comprised of midpoints of segments in F2. Since R lies on each of these segments, the midpoint of each segment must lie on the line parallel to set E1 passing through R. This is the line parallel to the perpendicular bisector of PQ going through R.

- Check the given points against E1

Check whether the given points lie on the perpendicular bisector of segment PQ. For point (-2,7), it coincides with point P, where S1 is tangent, so it lies in E1. For point (0,3/2), we use the equation of the perpendicular bisector obtained in Step 3 and check if it satisfies that equation. If it does not, then it does not lie in E1.

- Check the given points against E2

Use the equation of the line parallel to E1 through R to check if the points (4/5, 7/5) and (1/2, 1) lie on it. If they do, they are in E2. If not, they are outside of E2.

Key concepts

Tangent to Circles

Understanding the concept of a tangent to a circle is pivotal in solving numerous geometrical problems. In the context of the given problem, a tangent is a line that touches a circle at precisely one point. This unique point is significant because it's the only point where the tangent and the circle meet without cutting through the circle.

For a line to be a tangent to a circle, it must satisfy the condition of being perpendicular to the radius of the circle at the point of contact. This scenario is illustrated in the exercise where line T is tangent to circles S1 and S2 at points P and Q, respectively. Consequently, the radii from the centre of circles S1 and S2 to points P and Q form right angles with line T. Such perpendicular connections ensure that no matter how large or small circles S1 and S2 become, as long as they remain tangent to line T at the designated points, they will abide by this fundamental property of tangency.

Locus of a Point

The locus of a point refers to the set of all positions that a point can occupy, satisfying a given condition or set of conditions. When a problem talks about the locus, it’s essentially describing a path or a line that the point follows based on the conditions imposed. In the exercise, 'set E1' can be translated as the locus of point M. The locus here is the path that point M traces as the circles S1 and S2, tangent at P and Q respectively, touch each other.

As the problem dictates M to be the point where the two circles meet, M’s position isn't random but follows a precise path - the perpendicular bisector of the line segment PQ. This locus emerges because, by definition, the tangent at a point of a circle is equidistant from the ends of the diameter through the point of tangency. Consequently, as circles S1 and S2 alter in size while keeping the tangency condition at P and Q, M's locus, by definition, can only be the line equally distant from P and Q - the very definition of a perpendicular bisector.

Line Segments and Mid-Points

Line segments are part of lines bounded by two endpoints. The mid-point of a line segment is the point that divides the segment into two equal lengths. The significance of mid-points is often related to symmetry and balance within geometrical shapes and designs.

In the context of JEE Advanced Mathematics problems like the presented one, calculating the mid-point of a line segment involves finding the average of the x-coordinates and y-coordinates of the endpoints respectively. For 'set F2', which contains line segments passing through point R (1,1), this property of mid-points plays a crucial role. A line segment's midpoint represents balance, and when numerous line segments pass through a common point as in 'set F2', their mid-points will form a geometric locus on their own.

In set E2, these mid-points construct a set with a specific geometric property: they lie on a parallel line to the locus of M (E1) passing through R. This is because if R is a point that each of the line segments must pass through, the midpoint of each segment is the balance point between the segment's endpoints and thus reflects the symmetry about R.