The locus of the mid-point of the chord of co | vedclass

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

Question

(A) Let $P(t, \frac{4t - 20}{5})$ be a point on the line $4x - 5y = 20$.The equation of the chord of contact of tangents drawn from $P(t, \frac{4t - 2

Vedclass · Accepted Answer

$20(x^2 + y^2) - 36x + 45y = 0$

Vedclass · Answer

(A) Let $P(t, \frac{4t - 20}{5})$ be a point on the line $4x - 5y = 20$.The equation of the chord of contact of tangents drawn from $P(t, \frac{4t - 20}{5})$ to the circle $x^2 + y^2 = 9$ is given by $T = 0$:$tx + (\frac{4t - 20}{5})y = 9$  --- $(1)$Let $M(h, k)$ be the mid-point of this chord. The equation of a chord of a circle with mid-point $(h, k)$ is given by $T = S_1$:$hx + ky = h^2 + k^2$  --- $(2)$Comparing equations $(1)$ and $(2)$,we have:$\frac{t}{h} = \frac{4t - 20}{5k} = \frac{9}{h^2 + k^2}$From $\frac{t}{h} = \frac{9}{h^2 + k^2}$,we get $t = \frac{9h}{h^2 + k^2}$.From $\frac{4t - 20}{5k} = \frac{9}{h^2 + k^2}$,we get $4t - 20 = \frac{45k}{h^2 + k^2}$.Substituting $t$ in the second equation:$4(\frac{9h}{h^2 + k^2}) - 20 = \frac{45k}{h^2 + k^2}$$36h - 20(h^2 + k^2) = 45k$$20(h^2 + k^2) - 36h + 45k = 0$Replacing $(h, k)$ with $(x, y)$,the locus is:$20(x^2 + y^2) - 36x + 45y = 0$

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is

Similar Questions

Let $S$ be the focus of the parabola $y^2=8x$ and let $PQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the triangle $PQS$ is:

If the line $x+y+1=0$ intersects the circle $x^2+y^2+x+3y=0$ at two points $A$ and $B$,then the centre of the circle which passes through the points $A, B$ and the point of intersection of the tangents drawn at $A$ and $B$ to the given circle is

If the line $x - 2y = k$ cuts off a chord of length $2$ from the circle ${x^2} + {y^2} = 3$,then $k =$

The length of the common chord of the circles $x^2+y^2+3x+5y+4=0$ and $x^2+y^2+5x+3y+4=0$ is

The equation of the circle whose diameter is the common chord of the circles $x^2+y^2+2x+2y+1=0$ and $x^2+y^2+4x+6y+4=0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module