If the lengths of tangents drawn from a point | vedclass

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

Question

(C) Let $P(x_1, y_1)$ be the point. The length of the tangent from $P$ to a circle $S=0$ is $\sqrt{S}$.For the given circles,we normalize them to the

Vedclass · Accepted Answer

$\left(8, -\frac{15}{2}\right)$

Vedclass · Answer

(C) Let $P(x_1, y_1)$ be the point. The length of the tangent from $P$ to a circle $S=0$ is $\sqrt{S}$.For the given circles,we normalize them to the form $x^2+y^2+2gx+2fy+c=0$:$S_1: x^2+y^2-8x+40=0$$S_2: x^2+y^2-5x+16=0$$S_3: x^2+y^2-8x+16y+160=0$Since the lengths of the tangents are equal,$S_1 = S_2 = S_3$ at point $P(x_1, y_1)$:$x_1^2+y_1^2-8x_1+40 = x_1^2+y_1^2-5x_1+16 = x_1^2+y_1^2-8x_1+16y_1+160$Equating $S_1 = S_3$:$-8x_1+40 = -8x_1+16y_1+160$ $\Rightarrow 16y_1 = -120$ $\Rightarrow y_1 = -\frac{15}{2}$Equating $S_1 = S_2$:$-8x_1+40 = -5x_1+16$ $\Rightarrow 3x_1 = 24$ $\Rightarrow x_1 = 8$Thus,the point $P$ is $\left(8, -\frac{15}{2}\right)$.

If the lengths of tangents drawn from a point $P$ to the circles $x^2+y^2-8x+40=0$,$5x^2+5y^2-25x+80=0$,and $x^2+y^2-8x+16y+160=0$ are equal,then the point $P$ is:

Similar Questions

If $(a, b)$ and $(c, d)$ are the internal and external centres of similitude of the circles $x^2+y^2+4x-5=0$ and $x^2+y^2-6y+8=0$ respectively,then $(a+d)(b+c)=$

The equation of a circle passing through the points of intersection of the circles $x^2 + y^2 + 13x - 3y = 0$ and $2x^2 + 2y^2 + 4x - 7y - 25 = 0$ and the point $(1, 1)$ is

If the circles $x^2+y^2-8x-8y+28=0$ and $x^2+y^2-8x-6y+25-\alpha^2=0$ have only one common tangent,then $\alpha=$

If the angle between the circles $x^2+y^2-4x-6y-3=0$ and $x^2+y^2+8x-4y+\lambda=0$ is $60^{\circ}$,then a value of $\lambda$ is

If $(p, q)$ is the centre of the circle which cuts the three circles $x^2+y^2-2x-4y+4=0$,$x^2+y^2+2x-4y+1=0$ and $x^2+y^2-4x-2y-11=0$ orthogonally,then $p+q=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module