SATHEE: Circle 2 Question 4

Circle 2 Question 4

5. If the circles $x^{2} + y^{2} - 16 x - 20 y + 164 = r^{2}$ and $(x - 4)^{2} + (y - 7)^{2} = 36$ intersect at two distinct points, then

(2019 Main, 9 Jan II)

(a) $0 < r < 1$

(b) $r > 11$

(d) $r = 11$

Show Answer

Answer:

Correct Answer: 5. (c)

Solution:

Circle I is $x^{2} + y^{2} - 16 x - 20 y + 164 = r^{2}$

$\Rightarrow (x - 8)^{2} + (y - 10)^{2} = r^{2}$

$\Rightarrow C_{1} (8, 10)$ is the centre of Istcircle and $r_{1} = r$ is its radius Circle II is $(x - 4)^{2} + (y - 7)^{2} = 36$

$\Rightarrow C_{2} (4, 7)$ is the centre of 2 nd circle and $r_{2} = 6$ is its radius.

Two circles intersect if $| r_{1} - r_{2} | < C_{1} C_{2} < r_{1} + r_{2}$

$\Rightarrow | r - 6 | < \sqrt{(8 - 4)^{2} + (10 - 7)^{2}} < r + 6$

$\Rightarrow | r - 6 | < \sqrt{16 + 9} < r + 6$

$\Rightarrow | r - 6 | < 5 < r + 6$

Now as, $5 < r + 6$ always, we have to solve only

$| r - 6 | < 5 \Rightarrow - 5 < r - 6 < 5$

$\Rightarrow 6 - 5 < r < 5 + 6 \Rightarrow 1 < r < 11$

Circle 2 Question 4

5. If the circles $x^{2} + y^{2} - 16 x - 20 y + 164 = r^{2}$ and $(x - 4)^{2} + (y - 7)^{2} = 36$ intersect at two distinct points, then

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Circle 2 Question 4

5. If the circles x2+y2−16x−20y+164=r2 and (x−4)2+(y−7)2=36 intersect at two distinct points, then

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

5. If the circles $x^{2} + y^{2} - 16 x - 20 y + 164 = r^{2}$ and $(x - 4)^{2} + (y - 7)^{2} = 36$ intersect at two distinct points, then