If a tangent to the circle x^2 + y^2 = 1 inte | vedclass

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

Question

(C) Let the mid-point of $PQ$ be $S(h, k).$Since $P$ lies on the $x$-axis and $Q$ lies on the $y$-axis,let $P = (a, 0)$ and $Q = (0, b).$The mid-point

Vedclass · Accepted Answer

$x^2 + y^2 - 4x^2y^2 = 0$

Vedclass · Answer

(C) Let the mid-point of $PQ$ be $S(h, k).$Since $P$ lies on the $x$-axis and $Q$ lies on the $y$-axis,let $P = (a, 0)$ and $Q = (0, b).$The mid-point $S(h, k)$ is given by $h = \frac{a}{2}$ and $k = \frac{b}{2},$ so $a = 2h$ and $b = 2k.$The equation of the line $PQ$ is $\frac{x}{a} + \frac{y}{b} = 1,$ which becomes $\frac{x}{2h} + \frac{y}{2k} = 1.$Since this line is a tangent to the circle $x^2 + y^2 = 1,$ the perpendicular distance from the origin $(0, 0)$ to the line must be equal to the radius $r = 1.$The distance $d = \frac{|-1|}{\sqrt{(\frac{1}{2h})^2 + (\frac{1}{2k})^2}} = 1.$Squaring both sides,we get $\frac{1}{\frac{1}{4h^2} + \frac{1}{4k^2}} = 1,$which simplifies to $\frac{1}{4h^2} + \frac{1}{4k^2} = 1.$Multiplying by $4h^2k^2,$ we get $k^2 + h^2 = 4h^2k^2.$Replacing $(h, k)$ with $(x, y),$ the locus is $x^2 + y^2 = 4x^2y^2$ or $x^2 + y^2 - 4x^2y^2 = 0.$

If a tangent to the circle $x^2 + y^2 = 1$ intersects the coordinate axes at distinct points $P$ and $Q,$ then the locus of the mid-point of $PQ$ is

Similar Questions

Let a point $P$ be such that its distance from the point $(5, 0)$ is thrice the distance of $P$ from the point $(-5, 0)$. If the locus of the point $P$ is a circle of radius $r$,then $4r^{2}$ is equal to ...... .

The equation of the locus of the centroid of the triangle whose vertices are $(a \cos k, a \sin k)$,$(b \sin k, -b \cos k)$ and $(1, 0)$,where $k$ is a parameter,is

If a circle of constant radius $3k$ passes through the origin $O$ and meets the coordinate axes at $A$ and $B$,then the locus of the centroid of the triangle $OAB$ is

The centres of those circles which touch the circle $x^{2} + y^{2} - 8x - 8y - 4 = 0$ externally and also touch the $x$-axis,lie on:

$A$ rod $PQ$ of length $2a$ moves with its ends on the coordinate axes. Find the locus of the circumcenter of $\Delta OPQ$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module