Tangents are drawn to a unit circle with cent | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the point on the line be $P(x_1, y_1)$. Since $P$ lies on the line $2x + y = 4$,we have $2x_1 + y_1 = 4$ ... $(1)$.The equation of the chord o

Vedclass · Accepted Answer

$4 (x^2 + y^2) = 2x + y$

Vedclass · Answer

(C) Let the point on the line be $P(x_1, y_1)$. Since $P$ lies on the line $2x + y = 4$,we have $2x_1 + y_1 = 4$ ... $(1)$.The equation of the chord of contact of tangents drawn from $P(x_1, y_1)$ to the unit circle $x^2 + y^2 = 1$ is $xx_1 + yy_1 = 1$.Let $(h, k)$ be the midpoint of this chord. The equation of a chord with midpoint $(h, k)$ for the circle $x^2 + y^2 = 1$ is $xh + yk = h^2 + k^2$.Comparing the two equations of the same chord:$\frac{x_1}{h} = \frac{y_1}{k} = \frac{1}{h^2 + k^2}$Thus,$x_1 = \frac{h}{h^2 + k^2}$ and $y_1 = \frac{k}{h^2 + k^2}$.Substituting these into equation $(1)$:$2 \left( \frac{h}{h^2 + k^2} \right) + \frac{k}{h^2 + k^2} = 4$$2h + k = 4(h^2 + k^2)$Replacing $(h, k)$ with $(x, y)$,the locus is $4(x^2 + y^2) = 2x + y$.

Tangents are drawn to a unit circle with centre at the origin from each point on the line $2x + y = 4$. Then the equation to the locus of the middle point of the chord of contact is

Similar Questions

If the distances from the origin to the centres of the three circles $x^2 + y^2 - 2\lambda_i x = c^2$ for $i = 1, 2, 3$ are in $G.P.$,then the lengths of the tangents drawn to them from any point on the circle $x^2 + y^2 = c^2$ are in

The tangent at $A(-1, 2)$ on the circle $x^2+y^2-4x-8y+7=0$ touches the circle $x^2+y^2+4x+6y=0$ at $B$. Then,a point of trisection of $AB$ is

The coordinates of the point on the circle $x^2 + y^2 - 12x - 4y + 30 = 0$ which is farthest from the origin are:

If $\theta$ is the angle subtended at $P(x_1, y_1)$ by the circle $S \equiv x^2 + y^2 + 2gx + 2fy + c = 0$,then

If two circles of the same radius $a$ and centers at $(2, 3)$ and $(5, 6)$ are orthogonal,find the value of $a$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module