If the tangent to ellipse x2+2 y=1 at point P((1/√2), (1/2)) meets the auxiliary circle at the points R and Q, then tangents to circle at Q and R intersect at

Question

If the tangent to ellipse x2+2 y=1 at point P((1/√2), (1/2)) meets the auxiliary circle at the points R and Q, then tangents to circle at Q and R intersect at (A)  ( (1/√2), 1 )  (B)  ( 1, (1/√2) )  (C)  ( (1/2), (1/2) )  (D)  ( (1/2), (1/√2) )

Tardigrade Admin · Accepted Answer

Equation of tangent to ellipse at given point is

x (\frac{1}{2}) + 2 y (\frac{1}{2}) = 1

\Rightarrow x + 2 y = 2

... (i)

Now,

QR

is chord of contact of circle

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

with respect to point

T (h, K)

.
then,

QR \equiv h x + Ky = 1

... (ii)
Equations (I) and (li) represent same straight line, then comparing ratio of coefficients we have

\frac{h}{1} = \frac{K}{2} = \frac{1}{2}

Hence;

(h, K) \equiv (\frac{1}{2}, 1)

Q. If the tangent to ellipse $x^{2} + 2 y = 1$ at point $P (\frac{1}{2}, \frac{1}{2})$ meets the auxiliary circle at the points $R$ and $Q$ , then tangents to circle at $Q$ and $R$ intersect at

$(\frac{1}{2}, 1)$

$(1, \frac{1}{2})$

$(\frac{1}{2}, \frac{1}{2})$

$(\frac{1}{2}, \frac{1}{2})$

Q. If the tangent to ellipse x2+2y=1 at point P(2​1​,21​) meets the auxiliary circle at the points R and Q, then tangents to circle at Q and R intersect at

(2​1​,1)

(1,2​1​)

(21​,21​)

(21​,2​1​)

Q. If the tangent to ellipse $x^{2} + 2 y = 1$ at point $P (\frac{1}{2}, \frac{1}{2})$ meets the auxiliary circle at the points $R$ and $Q$ , then tangents to circle at $Q$ and $R$ intersect at

$(\frac{1}{2}, 1)$

$(1, \frac{1}{2})$

$(\frac{1}{2}, \frac{1}{2})$

$(\frac{1}{2}, \frac{1}{2})$