PARAGRAPH "X" Let S be the circle in the xy-plane defined by the equation x2 + y2 = 4. Question: Let P be a point on the circle S with both coordinates being positive. Let the tangent to S to P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve

Question

PARAGRAPH "X" Let S be the circle in the xy-plane defined by the equation x2 + y2 = 4. Question: Let P be a point on the circle S with both coordinates being positive. Let the tangent to S to P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve (A) (x + y)2 = 3xy  (B) x2/3 + y2/3 = 24/3 (C) x2 + y2 = 2xy  (D) x2 + y2 = x2 y2

Tardigrade Admin · Accepted Answer

Tangent at P(2 cos θ, 2 sin θ) is x cos θ + y sin θ = 2 M(2 sec θ , 0) and N(0, 2 cosec θ) Let midpoint be (h, k) h = sec θ , k = cosec θ (1/h2) + (1/k2) = 1 (1/x2) + (1/y2) = 1

$(x + y)^{2} = 3 x y$

$x^{2/3} + y^{2/3} = 2^{4/3}$

$x^{2} + y^{2} = 2 x y$

$x^{2} + y^{2} = x^{2} y^{2}$

Tangent at $P (2 cos θ, 2 sin θ)$ is $x cos θ + y sin θ = 2$
$M (2 sec θ, 0)$ and $N (0, 2 cosec θ)$
Let midpoint be (h, k)
$h = sec θ, k = cosec θ$
$\frac{1}{h ^{2}} + \frac{1}{k ^{2}} = 1$
$\frac{1}{x ^{2}} + \frac{1}{y ^{2}} = 1$

(x+y)2=3xy

x2/3+y2/3=24/3

x2+y2=2xy

x2+y2=x2y2

Tangent at P(2cosθ,2sinθ) is xcosθ+ysinθ=2 M(2secθ,0) and N(0,2cosecθ) Let midpoint be (h, k) h=secθ,k=cosecθ h21​+k21​=1 x21​+y21​=1

$(x + y)^{2} = 3 x y$

$x^{2/3} + y^{2/3} = 2^{4/3}$

$x^{2} + y^{2} = 2 x y$

$x^{2} + y^{2} = x^{2} y^{2}$

Tangent at $P (2 cos θ, 2 sin θ)$ is $x cos θ + y sin θ = 2$
$M (2 sec θ, 0)$ and $N (0, 2 cosec θ)$
Let midpoint be (h, k)
$h = sec θ, k = cosec θ$
$\frac{1}{h ^{2}} + \frac{1}{k ^{2}} = 1$
$\frac{1}{x ^{2}} + \frac{1}{y ^{2}} = 1$