Paragraph: Let S be the circle in the xy – plane defined by the...

Paragraph: Let

S

be the circle in the

xy

– plane defined by the equation

{x}^{2} + {y}^{2} = 4 .

Question : Let

P

be a point on the circle

S

with both coordinates being positive. Let the tangent to

S

P

intersect the coordinate axes at the points

M

and

MN

Then, the mid-point of the line segment must lie on the curve Paragraph: Let S be the circle in the xy – plane defined by the equation x ^ 2 + y ^ 2 = 4 . Question : Let P be a

Paragraph: Let S be the circle in the xy – plane defined by the equation x ^ 2 + y ^ 2 = 4 . Question : Let P be a

Options

A ${\left(x + y\right)}^{2} = 3xy$
B ${x}^{\dfrac{2}{3}} + {y}^{\dfrac{2}{3}} = {2}^{\dfrac{4}{3}}$
C ${x}^{2} + {y}^{2} = 2xy$
D ${x}^{2} + {y}^{2} = {x}^{2}{y}^{2}$

Correct answer

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

Step-by-step solution

Tangent at

P\left(2cos \theta, 2sin \theta\right)isxcos \theta + ysin \theta = 2

M\left(2sec \theta, 0\right)andN\left(0,2cosce \theta\right)

Let midpoint be (h, k)

h = sec \theta, k = cosce \theta

\dfrac{1}{{h}^{2}} + \dfrac{1}{{k}^{2}} = 1

\dfrac{1}{{x}^{2}} + \dfrac{1}{{y}^{2}} = 1

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