JEE Advance - Mathematics (2018 - Paper 2 Offline - No. 2)
Let T be the line passing through the points P($$-$$2, 7) and Q(2, $$-$$5). Let F1 be the set of al pairs of circles (S1, S2) such that T is tangent to S1 at P and tangent to S2 at Q, and also such that S1 and S2 touch each other at a point, say M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1. Let the set of all straight line segments joining a pair of distinct points of E1 and passing through the point R(1, 1) be F2. Let E2 be the set of the mid-points of the line segments in the set F2. Then, which of the following statement(s) is (are) TRUE?
The point ($$-$$2, 7) lies in E1
The point $$\left( {{4 \over 5},{7 \over 5}} \right)$$ does not lie in E2
The point $$\left( {{1 \over 2},1} \right)$$ lies in E2
The point $$\left( {0,{3 \over 2}} \right)$$ does not lie in E1
Explanation
It is given that T is tangents to S1 at P and S2 at Q and S1 and S2 touch externally at M.
$$ \therefore $$ MN = NP = NQ
$$ \therefore $$ Locus of M is a circle having PQ as its diameter of circle
$$ \therefore $$ Equation of circle
$$(x - 2)(x + 2) + (y + 5)(y - 7) = 0$$
$$ \Rightarrow {x^2} + {y^2} - 2y - 39 = 0$$
Hence,
$${E_1}:{x^2} + {y^2} - 2y - 39 = 0$$, $$x \ne \pm 2$$
Locus of mid-point of chord (h, k) of the circle E1 is
$$xh + yk - (y + k) - 39 = {h^2} + {k^2} - 2k - 39$$
$$ \Rightarrow xh + yk - y - k = {h^2} + {k^2} - 2k$$
Since, chord is passing through (1, 1).
$$ \therefore $$ Locus of mid-point of chord (h, k) is
$$h + k - 1 - k = {h^2} + {k^2} - 2k$$
$$ \Rightarrow {h^2} + {k^2} - 2k - h + 1 = 0$$
Locus is $${E_2}:{x^2} + {y^2} - x - 2y + 1 = 0$$
Now, after checking options, (a) and (d) are correct.
$$ \therefore $$ MN = NP = NQ
$$ \therefore $$ Locus of M is a circle having PQ as its diameter of circle
$$ \therefore $$ Equation of circle
$$(x - 2)(x + 2) + (y + 5)(y - 7) = 0$$
$$ \Rightarrow {x^2} + {y^2} - 2y - 39 = 0$$
Hence,
$${E_1}:{x^2} + {y^2} - 2y - 39 = 0$$, $$x \ne \pm 2$$
Locus of mid-point of chord (h, k) of the circle E1 is
$$xh + yk - (y + k) - 39 = {h^2} + {k^2} - 2k - 39$$
$$ \Rightarrow xh + yk - y - k = {h^2} + {k^2} - 2k$$
Since, chord is passing through (1, 1).
$$ \therefore $$ Locus of mid-point of chord (h, k) is
$$h + k - 1 - k = {h^2} + {k^2} - 2k$$
$$ \Rightarrow {h^2} + {k^2} - 2k - h + 1 = 0$$
Locus is $${E_2}:{x^2} + {y^2} - x - 2y + 1 = 0$$
Now, after checking options, (a) and (d) are correct.
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