JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Straight Lines

Let

A (1, 4)

and

B (1, - 5)

be two points. Let

P

be a point on the circle

(x - 1)^{2} + (y - 1)^{2} = 1

such that

(P A)^{2} + (P B)^{2}

have maximum value, then the points

P, A

and

B

lie on :

(A)an ellipse
(B)a straight line✓ Correct
(C)a parabola
(D)a hyperbola

Answer: (B)

Solution

Write

P = (1 + cos θ, 1 + sin θ)

for a point of the circle of centre

(1, 1)

and radius

1

. Then

P A^{2} = cos^{2} θ + (sin θ - 3)^{2} = 10 - 6 sin θ, P B^{2} = cos^{2} θ + (sin θ + 6)^{2} = 37 + 12 sin θ

P A^{2} + P B^{2} = 47 + 6 sin θ

, which is greatest when

sin θ = 1

, that is at

P = (1, 2)

.
The abscissa of

(1, 2)

(1, 4)

and

(1, - 5)

is the same, so

P

A

and

B

all lie on the straight line

x = 1

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Single correct option

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