Question

If the centre of a circle is (2a, a-7), then the values of 'a', if the circle passes through the point (11,-9) and has diameter $10\sqrt{2}$ units will be:

• A. 5 and 2
• B. 5 and 3
• C. 3 and 4
• D. 4 and 5

Answer 1

The correct option is B

5 and 3

By given condition,

Distance between the centre C(2a, a-7) and the point P(11,-9), which lie on the circle = Radius of the circle
$\therefore$ Radius of circle = $\sqrt{(11-2a{)}^2+(-9-a+7{)}^2}...\left(i\right)$
$\because$ Distance between the points $(x_1,y_1)and(x_2,y_2)$
$d=\sqrt{(x_2-x_1{)}^2+(x_2-y_1{)}^2}$

Given that, length of diameter = $10\sqrt{2}$
Length of radius = $\frac{Lengthofdiameter}{2}$
$=\frac{10\sqrt{2}}{2}=5\sqrt{2}$
Putting this value in equation (i), we get
$50=(11-2a{)}^2+(2+a{)}^2\Rightarrow 50=121+4a^2-44a+4+a^2+4a\Rightarrow 5a^2-40a+75=0\Rightarrow a^2-8a+15=0\Rightarrow a^2-5a-3a+15=0$ [by factorization method]
$\Rightarrow a(a-5)-3(a-5)=0\Rightarrow (a-5)(a-3)=0\therefore a=3,5$
Hence, the required values of 'a' are 5 and 3.