If the curves, x2 – 6x + y2 + 8 = 0 and
x2 – 8y + y2 + 16 – k = 0, (k > 0) touch each other
at a point, then the largest value of k is ______.
Answer (integer)
36
Solution
C<sub>1</sub> : x<sup>2</sup> + y<sup>2</sup> – 6x + + 8 = 0
<br><br>C<sub>1</sub>(3, 0) and r<sub>1</sub> = 1
<br><br>C<sub>2</sub> : x<sup>2</sup> + y<sup>2</sup> – 8y + 16 – k = 0
<br><br>C<sub>2</sub>(0, 4) and r<sub>2</sub> = $\sqrt k$
<br><br>Two circles touch each other
<br><br>$\therefore$ C<sub>1</sub>C<sub>2</sub> = | r<sub>1</sub> $\pm$ r<sub>2</sub> |
<br><br>$\Rightarrow$ 5 = | 1 $\pm$ $\sqrt k$ |
<br><br>$\therefore$ 1 + $\sqrt k$ = 5 or $\sqrt k$ - 1 = 5
<br><br>$\Rightarrow$ k = 16 or k = 36
<br><br>So largest value of k = 36.
About this question
Subject: Mathematics · Chapter: Circles · Topic: Equation of a Circle
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