Any circle passes through the point of inters | vedclass

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

Question

(A) Let the point of intersection of the two lines be $A$.The angle subtended by the arc $PQ$ at the centre $C$ is twice the angle subtended by the ch

Vedclass · Accepted Answer

$180$

Vedclass · Answer

(A) Let the point of intersection of the two lines be $A$.The angle subtended by the arc $PQ$ at the centre $C$ is twice the angle subtended by the chord $PQ$ at any point on the circumference (in the alternate segment).For the line $x + \sqrt{3}y = 1$,the slope is $m_1 = -\frac{1}{\sqrt{3}}$.For the line $\sqrt{3}x - y = 2$,the slope is $m_2 = \sqrt{3}$.Since $m_1 	imes m_2 = (-\frac{1}{\sqrt{3}}) 	imes \sqrt{3} = -1$,the two lines are perpendicular to each other.Therefore,the angle between the lines at their intersection point $A$ is $90^o$.Since the lines are perpendicular,the chord $PQ$ subtends an angle of $90^o$ at the circumference point $A$. By the circle theorem,the angle subtended by the arc $PQ$ at the centre is $2 	imes 90^o = 180^o$.Thus,the chord $PQ$ is a diameter of the circle.

Any circle passes through the point of intersection of the lines $x + \sqrt{3}y = 1$ and $\sqrt{3}x - y = 2$. If it intersects these lines at points $P$ and $Q$,then the angle subtended by the arc $PQ$ at its centre is ............ $^o$.

Similar Questions

If the two circles $2x^2 + 2y^2 - 3x + 6y + k = 0$ and $x^2 + y^2 - 4x + 10y + 16 = 0$ intersect orthogonally,then the value of $k$ is:

The line $x+y+2=0$ intersects the circle $x^2+y^2+4x-4y-4=0$ at two points $A$ and $B$. Let $S \equiv x^2+y^2+2gx+2fy+c=0$ be a different circle passing through the points $A$ and $B$. If the distance of the centre of $S=0$ from $AB$ is $\sqrt{2}$,then $g+f+c=$

The equation of a circle that intersects the circle $x^2 + y^2 + 14x + 6y + 2 = 0$ orthogonally and whose centre is $(0, 2)$ is

The equation of the radical axis of the circles $x^2 + y^2 + x - y + 2 = 0$ and $3x^2 + 3y^2 - 4x - 12 = 0$ is:

The gradient of the radical axis of the circles ${x^2} + {y^2} - 3x - 4y + 5 = 0$ and $3{x^2} + 3{y^2} - 7x + 8y + 11 = 0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module