You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Equation of Tangent in Parametric Form

Let S be the ...

Question

Let $S$ be the circle in the $x y$ -plane defined by the equation $x^{2} + y^{2} = 4$ .

Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$ . Then, the mid-point of the line segment $M N$ must lie on the curve

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D $x^{2} + y^{2} = x^{2} y^{2}$
Given circle is $x^{2} + y^{2} = 4$
Assuming $P = (2 cos θ, 2 sin θ)$

Let the midpoint of $M N = (h, k)$
Tangent at $P$ ,
$2 cos θ \cdot x + 2 sin θ \cdot y - 4 = 0 \Rightarrow x cos θ + y sin θ - 2 = 0$
So, the coordinates of $M$ and $N$ are,
$M = (\frac{2}{cos θ}, 0) N = (0, \frac{2}{sin θ})$

Now, the midpoint is
$(h, k) = (\frac{1}{cos θ}, \frac{1}{sin θ}) \Rightarrow \frac{1}{h^{2}} + \frac{1}{k^{2}} = 1$

Hence, the midpoint of $M N$ lies on the curve
$x^{2} + y^{2} = x^{2} y^{2}$

Suggest Corrections

Similar questions

Q. Let

S

be the circle in the

x y

-plane defined by the equation

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

.

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

N

. Then, the mid-point of the line segment

M N

must lie on the curve

Q. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at 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

Q. Let

S

be the circle in the

x y

-plane defined by the equation

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

.

Let

E_{1} E_{2}

and

F_{1} F_{2}

be the chords of

S

passing through the point

P_{0} (1, 1)

and parallel to the

x

-axis and the

y

-axis, respectively. Let

G_{1} G_{2}

be the chord of

S

passing through

P_{0}

and having slope

- 1

. Let the tangents to

S

E_{1}

and

E_{2}

meet at

E_{3}

, the tangents to

S

F_{1}

and

F_{2}

meet at

F_{3}

, and the tangents to

S

G_{1}

and

G_{2}

meet at

G_{3}

. Then, the points

E_{3}, F_{3}

, and

G_{3}

lie on the curve

Q. Let S be the circle in the xy-plane defined by the equation

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

. Let

E_{1} E_{2}

and

F_{1} F_{2}

be the chord of S passing through the point

P_{o} (1,

and parallel to the x-axis and the y-axis, respectively. Let

G_{1} G_{2}

be the chord of S passing through

P_{o}

and having slop-

1

. Let the tangents to S at

E_{1}

and

E_{2}

meet at

E_{3}

, the tangents of S at

F_{1}

and

F_{2}

meet at

F_{3}

, and the tangents to S at

G_{1}

and

G_{2}

meet at

G_{3}

. Then, the points

E_{3}

F_{3}

and

G_{3}

lie on the curve.

Q. Let

R S

be the diameter of the circle

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

, where

S

is the point

(1, 0)

. Let

P

be a variable point (other than

R

and

S

) on the circle and tangents to the circle at

S

and

P

meet at the point

Q

. The normal to the circle at

P

intersects a line drawn through

Q

parallel to

R S

at point

E

. Then the locus of

E

passes through the point(s)

Join BYJU'S Learning Program

Explore more

Equation of Tangent in Parametric Form

Standard XII Mathematics

Join BYJU'S Learning Program

Let S be the circle in the xy-plane defined by the equation x2+y2=4. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at 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