If y = 2x is a chord of the circle x^2 + y^2 | vedclass

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

Question

(D) The family of circles passing through the intersection of the circle $S \equiv x^2 + y^2 - 10x = 0$ and the line $L \equiv y - 2x = 0$ is given by

Vedclass · Accepted Answer

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

Vedclass · Answer

(D) The family of circles passing through the intersection of the circle $S \equiv x^2 + y^2 - 10x = 0$ and the line $L \equiv y - 2x = 0$ is given by $S + \lambda L = 0$.$x^2 + y^2 - 10x + \lambda(y - 2x) = 0$$x^2 + y^2 - (10 + 2\lambda)x + \lambda y = 0$The center of this circle is $\left( \frac{10 + 2\lambda}{2}, -\frac{\lambda}{2} \right) = (5 + \lambda, -\frac{\lambda}{2})$.Since the chord $y = 2x$ is a diameter of this circle,the center must lie on the line $y = 2x$.$-\frac{\lambda}{2} = 2(5 + \lambda)$$-\frac{\lambda}{2} = 10 + 2\lambda$$-\lambda = 20 + 4\lambda$$-5\lambda = 20 \Rightarrow \lambda = -4$.Substituting $\lambda = -4$ into the equation $x^2 + y^2 - (10 + 2\lambda)x + \lambda y = 0$:$x^2 + y^2 - (10 - 8)x - 4y = 0$$x^2 + y^2 - 2x - 4y = 0$.

If $y = 2x$ is a chord of the circle $x^2 + y^2 - 10x = 0$,then the equation of the circle of which this chord is a diameter is:

Similar Questions

$A$ circle $C$ passing through the point $(1, 1)$ bisects the circumference of the circle $x^2+y^2-2x=0$. If $C$ is orthogonal to the circle $x^2+y^2+2y-3=0$,then the centre of the circle $C$ is

The equation of the circle which cuts the circles $x^2+y^2+4x-7=0$,$2x^2+2y^2+3x+5y-9=0$,and $x^2+y^2+y=0$ orthogonally is

Statement $(A) :$ If two circles $x^2 + y^2 + 2gx + 2fy = 0$ and $x^2 + y^2 + 2g'x + 2f'y = 0$ touch each other,then $f'g = fg'$.
Reason $(R) :$ Two circles touch each other if the line joining their centers is perpendicular to all possible common tangents.

If the lengths of the tangents drawn from a point $P$ to the three circles $x^2+y^2-4=0$,$x^2+y^2-2x+3y=0$,and $x^2+y^2+7y-18=0$ are equal,then the coordinates of $P$ are

The equation of the circle which intersects circles ${x^2} + {y^2} + x + 2y + 3 = 0$,${x^2} + {y^2} + 2x + 4y + 5 = 0$,and ${x^2} + {y^2} - 7x - 8y - 9 = 0$ at right angles is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module