P is a variable on the line y = 4. Tangents are drawn to the circle x2 + y2= 4 from P to touch it at A and B. The perpendicular PAQB is completed. Find the equation of the locus of Q. : Kaysons Education

Question

P is a variable on the line y = 4. Tangents are drawn to the circle x² + y²= 4 from P to touch it at A and B. The perpendicular PAQB is completed. Find the equation of the locus of Q.

diffcult

Solution

Correct option is

(x² + y²)(y + 4) = 2y

Let P (h, 4) be a variable point. Given circle is

x² + y² = 4 …(1)

Draw tangents from P (h, 4) and complete parallelogram PAQB.

Equation of the diagonal AB which is chord of contact of

x² + y²= 4 is

hx + 4y = 4 …(2)

Let co-ordinates of A and B are (x₁, y₁) and (x₂, y₂) respectively.

Since A (x₁, y₁) and B (x₂, y₂) lies on (2)

∴ hx₁ + 4y₁ = 4

and hx₂ + 4y₂ = 4

∴ h(x₁ + x₂) + 4(y₁ + y₂) = 8 …(3)

Since PAQB is parallelogram

∴ mid point of AB = mid point of PQ

Eliminating x from (1) and (2) then

From (3) and (5), we get

From (4) and (6)

from (4) and (6)

or (16 + h²)(α + h) = 8h …(8)

Dividing (8) by (7) then

Substituting the value h in (7) then

Hence locus of Q (α, β) is (x² + y²)(y + 4) = 2y²

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