Two circles of radii 4 text{ cm} and 1 text{ | vedclass

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

Question

(A) Let the two circles with centers $Q$ and $R$ and radii $r_1 = 4$ and $r_2 = 1$ touch each other externally. The distance between their centers is

Vedclass · Accepted Answer

$\frac{24}{25}$

Vedclass · Answer

(A) Let the two circles with centers $Q$ and $R$ and radii $r_1 = 4$ and $r_2 = 1$ touch each other externally. The distance between their centers is $QR = r_1 + r_2 = 4 + 1 = 5$.Let the direct common tangents meet at point $P$. The line joining the centers $Q$ and $R$ passes through $P$.Let $\alpha$ be the angle such that the total angle between the tangents is $2\alpha$. In the right-angled triangle formed by the center of the larger circle,the point of contact,and $P$,we have $\sin \alpha = \frac{r_1 - r_2}{QR} = \frac{4 - 1}{5} = \frac{3}{5}$.Since $\sin \alpha = \frac{3}{5}$,we have $\cos \alpha = \sqrt{1 - (\frac{3}{5})^2} = \frac{4}{5}$.The angle contained by the tangents is $	heta = 2\alpha$.Therefore,$\sin 	heta = \sin(2\alpha) = 2 \sin \alpha \cos \alpha = 2 	imes \frac{3}{5} 	imes \frac{4}{5} = \frac{24}{25}$.

Two circles of radii $4 \text{ cm}$ and $1 \text{ cm}$ touch each other externally and $\theta$ is the angle contained by their direct common tangents. Then $\sin \theta =$

Similar Questions

If $P$ is a point on the circle $x^{2}+y^{2}=4$, $Q$ is a point on the straight line $5x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of $PQ$, then $13$ times the sum of the abscissae of all such points $P$ is ........... .

Which equation of the chord bisects the circle $x^2 + y^2 = 8x$ at the point $(4, 3)$?

$A$ square is inscribed in the circle $x^2 + y^2 - 2x + 4y - 93 = 0$ with its sides parallel to the coordinate axes. The coordinates of its vertices are

The equation of the circumcircle of the triangle formed by the lines $y + \sqrt{3}x = 6$,$y - \sqrt{3}x = 6$,and $y = 0$ is:

The length of the diameter of a circle which touches the $x$-axis at the point $(1, 0)$ and passes through the point $(2, -3)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module