SATHEE: Circle 2 Question 7

Circle 2 Question 7

8. The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ is

(2015)

(a) 1

(b) 2

(d) 4

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Answer:

Correct Answer: 8. (c)

Solution:

PLAN Number of common tangents depend on the position of the circle with respect to each other.

(i) If circles touch externally $\Rightarrow C_{1} C_{2} = r_{1} + r_{2}, 3$ common tangents.

(ii) If circles touch internally $\Rightarrow C_{1} C_{2} = r_{2} - r_{1}, 1$ common tangent.

(iii) If circles do not touch each other, 4 common tangents.

Given equations of circles are

$\begin{array}{r} x^{2} + y^{2} - 4 x - 6 y - 12 = 0 \\ x^{2} + y^{2} + 6 x + 18 y + 26 = 0 \end{array}$

Centre of circle (i) is $C_{1} (2, 3)$ and radius

$= \sqrt{4 + 9 + 12} = 5 (r_{1})$

Centre of circle (ii) is $C_{2} (- 3, - 9)$ and radius

$= \sqrt{9 + 81 - 26} = 8 (r_{2})$

$\begin{aligned} Now, C_{1} C_{2} = \sqrt{(2 + 3)^{2} + (3 + 9)^{2}} \\ \Rightarrow C_{1} C_{2} = \sqrt{5^{2} + 12^{2}} \\ \Rightarrow C_{1} C_{2} = \sqrt{25 + 144} = 13 \\ ∴ r_{1} + r_{2} = 5 + 8 = 13 \\ Also, C_{1} C_{2} = r_{1} + r_{2} \end{aligned}$

Thus, both circles touch each other externally. Hence, there are three common tangents.

Circle 2 Question 7

8. The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ is

Answer:

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Circle 2 Question 7

8. The number of common tangents to the circles x2+y2−4x−6y−12=0 and x2+y2+6x+18y+26=0 is

Answer:

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8. The number of common tangents to the circles $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ and $x^{2} + y^{2} + 6 x + 18 y + 26 = 0$ is