Statement (A): The number of common tangents | vedclass

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

Question

(D) For circle $C_1: x^2 + y^2 = 4$,center $C_1 = (0, 0)$ and radius $r_1 = 2$.For circle $C_2: x^2 + y^2 - 6x - 8y - 24 = 0$,center $C_2 = (3, 4)$ an

Vedclass · Accepted Answer

$A$ is false but $R$ is true.

Vedclass · Answer

(D) For circle $C_1: x^2 + y^2 = 4$,center $C_1 = (0, 0)$ and radius $r_1 = 2$.For circle $C_2: x^2 + y^2 - 6x - 8y - 24 = 0$,center $C_2 = (3, 4)$ and radius $r_2 = \sqrt{3^2 + 4^2 - (-24)} = \sqrt{9 + 16 + 24} = \sqrt{49} = 7$.The distance between centers $|C_1C_2| = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = 5$.We check the condition $|C_1C_2|$ vs $r_1 + r_2$ and $|r_1 - r_2|$.$r_1 + r_2 = 2 + 7 = 9$.$|r_1 - r_2| = |2 - 7| = 5$.Since $|C_1C_2| = |r_1 - r_2| = 5$,the circles touch each other internally.When circles touch internally,they have only $1$ common tangent.Thus,Statement $(A)$ is false.Reason $(R)$ is a standard geometric theorem for circles,which is true.Therefore,$A$ is false but $R$ is true.

Statement $(A):$ The number of common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y = 24$ is $4$.
Reason $(R):$ For two circles with centers $C_1, C_2$ and radii $r_1, r_2$,if $|C_1C_2| > r_1 + r_2$,then the circles have $4$ common tangents.

Similar Questions

$A$ square is inscribed in the circle $x^2+y^2-2x+4y-93=0$ with its sides parallel to the coordinate axes. Which of the following can be one of the vertices of the square?

The curve $xy = c, (c > 0)$,and the circle $x^2 + y^2 = 1$ touch at two points. Then the distance between the points of contact is

$A$ line $l$ meets the circle $x^2+y^2=61$ at points $A$ and $B$. If $P(-5, 6)$ is a point such that $PA=PB=10$,then the equation of line $l$ is:

From a point $P$ on the line $4x - 3y = 6$,two tangents are drawn to the circle $x^2 + y^2 - 6x - 4y + 4 = 0$. If the angle between these tangents is $\tan^{-1}\left(\frac{24}{7}\right)$,then $P$ can be:

If the lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ are tangents to the same circle,then the area of that circle (in sq. units) is

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Statement $(A):$ The number of common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y = 24$ is $4$.Reason $(R):$ For two circles with centers $C_1, C_2$ and radii $r_1, r_2$,if $|C_1C_2| > r_1 + r_2$,then the circles have $4$ common tangents.