A and B are two points in xy-plane which are $2 \sqrt{2}$ unit distance apart and subtended an angle of 90⁰ at the point C(1, 2) on the line $x - y + 1 = 0$ which is larger than any angle subtended by the line segment AB at any other point on the line. The equation of the circle through the points A, B and C is ...........

see full answer

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

$x^{2} + y^{2} - 6 y + 7 = 0$

$x^{2} + y^{2} - 2 x - y + 3 = 0$

$x^{2} + y^{2} - 6 x + 6 y + 7 = 0$

$x^{2} + y^{2} - 4 x - 2 y + 3 = 0$

answer is A.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

$\bar{AB}$ subtends the greatest angle at C on the line $x - y + 1 = 0$ .

Therefore $x - y + 1 = 0$ touches the circle at C and hence $\bar{AB}$ as diameter .

required circle be $(x - 1)^{2} + (y - 2)^{2} + λ (x - y + 1) = 0$

$x^{2} + y^{2} + x (λ - 2) + y (- λ - 4) + (λ + 5) = 0$

$A B = 2 \sqrt{2} \Rightarrow radius = \sqrt{2}$

$\frac{{(λ - 2)}^{2}}{4} + \frac{{(λ + 4)}^{2}}{4} - (λ + 5) = 2 \Rightarrow 2 λ^{2} + 4 λ + 20 - 4 λ - 20 = 8 \Rightarrow λ^{2} = 4$

$\Rightarrow λ = \pm 2$

Equation of circle is $x^{2} + y^{2} - 6 y + 7 = 0$

Watch 3-min video & get full concept clarity