A point that lies on the common tangent to th | vedclass

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

Question

(D) Given circles are $S_1 \equiv x^2+y^2-2x+18y+78=0$ and $S_2 \equiv x^2+y^2+8x-6y-200=0$. Centre $C_1 = (1, -9)$ and radius $r_1 = \sqrt{1^2+(-9)^2

Vedclass · Accepted Answer

$\left(\frac{-2}{5}, \frac{-47}{4}\right)$

Vedclass · Answer

(D) Given circles are $S_1 \equiv x^2+y^2-2x+18y+78=0$ and $S_2 \equiv x^2+y^2+8x-6y-200=0$. Centre $C_1 = (1, -9)$ and radius $r_1 = \sqrt{1^2+(-9)^2-78} = \sqrt{1+81-78} = 2$. Centre $C_2 = (-4, 3)$ and radius $r_2 = \sqrt{(-4)^2+3^2-(-200)} = \sqrt{16+9+200} = 15$. Distance $C_1C_2 = \sqrt{(-4-1)^2+(3-(-9))^2} = \sqrt{(-5)^2+12^2} = \sqrt{25+144} = 13$. Since $r_2 - r_1 = 15 - 2 = 13 = C_1C_2$,the circles touch each other internally. The equation of the common tangent is $S_1 - S_2 = 0$. $(x^2+y^2-2x+18y+78) - (x^2+y^2+8x-6y-200) = 0$. $-10x + 24y + 278 = 0$,which simplifies to $-5x + 12y + 139 = 0$. Checking option $D$: $-5\left(\frac{-2}{5}\right) + 12\left(\frac{-47}{4}\right) + 139 = 2 - 141 + 139 = 0$. Thus,the point $\left(\frac{-2}{5}, \frac{-47}{4}\right)$ lies on the common tangent.

$A$ point that lies on the common tangent to the circles $x^2+y^2-2x+18y+78=0$ and $x^2+y^2+8x-6y-200=0$ among the following options is

Similar Questions

If the circles $x^{2}+y^{2}-2x-2y-7=0$ and $x^{2}+y^{2}+4x+2y+k=0$ cut orthogonally,then the length of the common chord of the circles is

The length of the common chord of the circles $x^2 + y^2 = 12$ and $x^2 + y^2 - 4x + 3y - 2 = 0$ is (in $\sqrt{2}$)

For the circles $(x-a)^2+y^2=a^2$ and $x^2+(y-a)^2=a^2$,where $a>0$,which one of the following is not true?

Tangents are drawn from the point $(4, 3)$ to the circle $x^2 + y^2 = 9$. The area of the triangle formed by them and the line joining their points of contact is

If $OA$ and $OB$ are the tangents to the circle $x^2 + y^2 - 6x - 8y + 21 = 0$ drawn from the origin $O$,then $AB =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module