SATHEE: Circle 4 Question 22

Circle 4 Question 22

22. Find the equations of the circles passing through $(- 4, 3)$ and touching the lines $x + y = 2$ and $x - y = 2$ .

$(1982, 3 M)$

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Answer:

Correct Answer: 22. $x^{2} + y^{2} + 2 (10 \pm 3 \sqrt{6}) x + (55 \pm 24 \sqrt{6}) = 0$

Solution:

Let the equation of the required circle be

$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

It passes through $(- 4, 3)$ .

$∴ 25 - 8 g + 6 f + c = 0$

Since, circle touches the line $x + y - 2 = 0$ and $x - y - 2 = 0$ .

$∴ \frac{- g - f - 2}{\sqrt{2}} = \frac{- g + f - 2}{\sqrt{2}} = \sqrt{g^{2} + f^{2} - c}$

$\begin{aligned} Now, & \frac{- g - f - 2}{\sqrt{2}} & = \frac{- g + f - 2}{\sqrt{2}} \\ \Rightarrow & - g - f - 2 & = \pm (- g + f - 2) \\ \Rightarrow & - g - f - 2 & = - g + f - 2 \\ or & - g - f - 2 & = g - f + 2 \\ \Rightarrow & f & = 0 or g = - 2 \end{aligned}$

Case I When $f = 0$

From Eq. (iii), we get

$\begin{aligned} \frac{- g - 2}{\sqrt{2}} & = \sqrt{g^{2} - c} \\ \Rightarrow g^{2} - 4 g - 4 - 2 c & = 0 \end{aligned}$

On putting $f = 0$ in Eq. (ii). we get

$25 - 8 g + c = 0$

Eliminating $c$ between Eqs. (iv) and (v), we get

$\Rightarrow \begin{aligned} g^{2} - 20 g + 46 & = 0 \\ g & = 10 \pm 3 \sqrt{6} and c = 55 \pm 24 \sqrt{6} \end{aligned}$

On substituting the values of $g, f$ and $c$ in Eq. (i), we get

$x^{2} + y^{2} + 2 (10 \pm 3 \sqrt{6}) x + (55 \pm 24 \sqrt{6}) = 0$

Case II When $g = - 2$

From Eq. (iii), we get

$\begin{array}{lc} \Rightarrow & f^{2} = 2 (4 + f^{2} - c) \Rightarrow & f^{2} - 2 c + 8 = 0 \end{array}$

On putting $g = - 2$ in Eq. (ii), we get

$c = - 6 f - 41$

On substituting $c$ in Eq. (vi), we get

$f^{2} + 12 f + 90 = 0$

This equation gives imaginary values of $f$ .

Thus, there is no circle in this case.

Hence, the required equations of the circles are

$x^{2} + y^{2} + 2 (10 \pm 3 \sqrt{6}) x + (55 \pm 24 \sqrt{6}) = 0$

Circle 4 Question 22

22. Find the equations of the circles passing through $(- 4, 3)$ and touching the lines $x + y = 2$ and $x - y = 2$ .

Answer:

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Circle 4 Question 22

22. Find the equations of the circles passing through (−4,3) and touching the lines x+y=2 and x−y=2.

Answer:

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22. Find the equations of the circles passing through $(- 4, 3)$ and touching the lines $x + y = 2$ and $x - y = 2$ .