Name: Two Circles Touching Internally and Externally
Uploaded: 2019-07-19
Duration: 6 min 6 s
Description: In this video, we will learn the condition for two circles touching internally and externally using an interesting example. Come, let's learn and watch this...

Question

Find the equation of circle passing through three points

(2, 1), (5, 5), (- 6, 7)

.

Accepted Answer

Let the circle is passing through the points $P (2, 1), Q (5, 5),$ and $R (- 6, 7)$ .

Let $C (a, b)$ be the centre of the required circle. Then, we have

$C P = C R$

$\Rightarrow C P^{2} = C R^{2}$

$\Rightarrow {(a - 2)}^{2} + {(b - 1)}^{2} = {(a + 6)}^{2} + {(b - 7)}^{2}$

$\Rightarrow a^{2} - 4 a + 4 + b^{2} - 2 b + 1 = a^{2} + 12 a + 36 + b^{2} - 14 b + 49$

$\Rightarrow - 4 a + 3 b = 20 \to (1)$

Also, $C P = C Q$

$\Rightarrow C P^{2} = C Q^{2}$

$\Rightarrow {(a - 2)}^{2} + {(b - 1)}^{2} = {(a - 5)}^{2} + {(b - 5)}^{2}$

$\Rightarrow a^{2} - 4 a + 4 + b^{2} - 2 b + 1 = a^{2} - 10 a + 25 + b^{2} - 10 b + 25$

$\Rightarrow 6 a + 8 b = 45 \to (2)$

Now, multiply equation $(1)$ by $6$ and multiply equation $(2)$ by $4$ then, add both the obtained equations, to get

$50 b = 300$

$\Rightarrow b = 6$

Now, substituting the value of $b$ in equation $(1)$ , we get

$- 4 a + 3 (6) = 20$

$\Rightarrow - 4 a = 2$

$\Rightarrow a = - \frac{1}{2}$

Hence, the centre of the required circle is $(- \frac{1}{2}, 6)$

Now, radius of the circle $= C P = \sqrt{{(- \frac{1}{2} - 2)}^{2} + {(6 - 1)}^{2}} = \sqrt{{(- \frac{5}{2})}^{2} + {(5)}^{2}} = \sqrt{\frac{125}{4}} = \frac{5 \sqrt{5}}{2}$

Hence, the equation of the required circle is ${(x + \frac{1}{2})}^{2} + {(y - 6)}^{2} = \frac{125}{4}$

$\Rightarrow x^{2} + x + \frac{1}{4} + y^{2} - 12 y + 36 = \frac{125}{4}$

$\Rightarrow x^{2} + y^{2} + x - 12 y + 5 = 0$ .

Find the equation of circle passing through three points $(2, 1), (5, 5), (- 6, 7)$ .

Important Questions on Circle

Learn and Prepare for any exam you want !

Find the equation of circle passing through three points (2,1),(5,5),(-6,7).

Important Questions on Circle

Learn and Prepare for any exam you want !

Find the equation of circle passing through three points $(2, 1), (5, 5), (- 6, 7)$ .