SATHEE: Circle 5 Question 21

Circle 5 Question 21

21. Lines $5 x + 12 y - 10 = 0$ and $5 x - 12 y - 40 = 0$ touch a circle $C_{1}$ of diameter 6 . If the centre of $C_{1}$ lies in the first quadrant, find the equation of the circle $C_{2}$ which is concentric with $C_{1}$ and cuts intercepts of length 8 on these lines.

(1986, 5M)

Show Answer

Answer:

Correct Answer: 21. $(x - 5)^{2} + (y - 2)^{2} = 5^{2}$

Solution:

Since, $5 x + 12 y - 10 = 0$ and $5 x - 12 y - 40 = 0$ are both perpendicular tangents to the circle $C_{1}$ .

$∴ O A B C$ forms a square.

Let the centre coordinates be $(h, k)$ , where,

$\begin{matrix} O C = O A = 3 and O B = 6 \sqrt{2} \\ \Rightarrow \frac{| 5 h + 12 k - 10 |}{13} = 3 and \frac{| 5 h + 12 k - 40 |}{13} = 3 \end{matrix}$

$\Rightarrow 5 h + 12 k - 10 = \pm 39$ and $5 h - 12 k - 40 = \pm 39$ on solving above equations. The coordinates which lie in I quadrant are $(5, 2)$ .

$∴$ Centre for $C_{1} (5, 2)$

To obtain equation of circle concentric with $C_{1}$ and making an intercept of length 8 on $5 x + 12 y = 10$ and $5 x - 12 y = 40$ $∴$ Required equation of circle $C_{2}$ has centre $(5, 2)$ and radius 5 is $(x - 5)^{2} + (y - 2)^{2} = 5^{2}$

Circle 5 Question 21

21. Lines $5 x + 12 y - 10 = 0$ and $5 x - 12 y - 40 = 0$ touch a circle $C_{1}$ of diameter 6 . If the centre of $C_{1}$ lies in the first quadrant, find the equation of the circle $C_{2}$ which is concentric with $C_{1}$ and cuts intercepts of length 8 on these lines.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Circle 5 Question 21

21. Lines 5x+12y−10=0 and 5x−12y−40=0 touch a circle C1 of diameter 6 . If the centre of C1 lies in the first quadrant, find the equation of the circle C2 which is concentric with C1 and cuts intercepts of length 8 on these lines.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

21. Lines $5 x + 12 y - 10 = 0$ and $5 x - 12 y - 40 = 0$ touch a circle $C_{1}$ of diameter 6 . If the centre of $C_{1}$ lies in the first quadrant, find the equation of the circle $C_{2}$ which is concentric with $C_{1}$ and cuts intercepts of length 8 on these lines.