JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Circle

A circle passing through the point

P (α, β)

in the first quadrant touches the two coordinate axes at the points A and B . The point P is above the line AB . The point Q on the line segment AB is the foot of perpendicular from P on AB . If PQ is equal to 11 units, then the value of

α β

is ____

Answer: 121

Solution

A circle touching both coordinate axes has its center at

(a, a)

for some

a > 0

, with radius

a

. Its equation is

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

.

Since P

(α, β)

lies on the circle:

(α - a)^{2} + (β - a)^{2} = a^{2}

α^{2} + β^{2} - 2 a α - 2 a β + a^{2} = 0 ...(1)

The line AB (joining the points of tangency) has equation

x + y = a

.

The perpendicular distance from P

(α, β)

to the line

x + y = a

is:

P Q = \frac{∣ α + β - a ∣}{2} = 11

∣ α + β - a ∣ = 112

(α + β - a)^{2} = 242

Expanding:

α^{2} + β^{2} + a^{2} + 2 α β - 2 a α - 2 a β = 242

From equation (1):

α^{2} + β^{2} - 2 a α - 2 a β + a^{2} = 0

, so:

2 α β = 242

α β = 121

JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Integer answer

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JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Circle

Solution

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