SATHEE: Circle 5 Question 3

Circle 5 Question 3

3. The centres of those circles which touch the circle, $x^{2} + y^{2} - 8 x - 8 y - 4 = 0$ , externally and also touch the $X$ -axis, lie on

(2016 Main)

(a) a circle

(b) an ellipse which is not a circle

(d) a parabola

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Answer:

Correct Answer: 3. (d)

Solution:

Given equation of circle is

$x^{2} + y^{2} - 8 x - 8 y - 4 = 0$ , whose centre is $C (4, 4)$ and radius $= \sqrt{4^{2} + 4^{2} + 4} = \sqrt{36} = 6$

Let the centre of required circle be $C_{1} (x, y)$ . Now, as it touch the $X$ -axis, therefore its radius $= | y |$ . Also, it touch the circle

$\begin{array}{lc} x^{2} + y^{2} - 8 x - 8 y - 4 = 0, therefore C C_{1} = 6 + | y | \\ \Rightarrow & \sqrt{(x - 4)^{2} + (y - 4)^{2}} = 6 + | y | \\ \Rightarrow & x^{2} + 16 - 8 x + y^{2} + 16 - 8 y = 36 + y^{2} + 12 | y | \\ \Rightarrow & x^{2} - 8 x - 8 y + 32 = 36 + 12 | y | \\ \Rightarrow & x^{2} - 8 x - 8 y - 4 = 12 | y | \end{array}$

Case I If $y > 0$ , then we have

$\begin{aligned} x^{2} - 8 x - 8 y - 4 & = 12 y \\ \Rightarrow & x^{2} - 8 x - 20 y - 4 & = 0 \\ \Rightarrow & x^{2} - 8 x - 4 & = 20 y \end{aligned}$

$\Rightarrow (x - 4)^{2} - 20 = 20 y$

$\Rightarrow (x - 4)^{2} = 20 (y + 1)$ which is a parabola.

Case II If $y < 0$ , then we have

$\begin{array}{lc} x^{2} - 8 x - 8 y - 4 = - 12 y \\ \Rightarrow & x^{2} - 8 x - 8 y - 4 + 12 y = 0 \\ \Rightarrow & x^{2} - 8 x + 4 y - 4 = 0 \\ \Rightarrow & x^{2} - 8 x - 4 = - 4 y \\ \Rightarrow & (x - 4)^{2} = 20 - 4 y \\ \Rightarrow & (x - 4)^{2} = - 4 (y - 5), which is again a parabola. \end{array}$

Circle 5 Question 3

3. The centres of those circles which touch the circle, $x^{2} + y^{2} - 8 x - 8 y - 4 = 0$ , externally and also touch the $X$ -axis, lie on

Answer:

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Circle 5 Question 3

3. The centres of those circles which touch the circle, x2+y2−8x−8y−4=0, externally and also touch the X-axis, lie on

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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3. The centres of those circles which touch the circle, $x^{2} + y^{2} - 8 x - 8 y - 4 = 0$ , externally and also touch the $X$ -axis, lie on