JEE Main 2021 (Online) 27th August Evening ShiftQuestion 1
Two circles each of radius $5$ units touch each other at the point $(1,2)$. If the equation of their common tangent is $4x+3y=10$, and $C_1(\alpha,\beta)$ and $C_2(\gamma,\delta)$, $C_1\neq C_2$ are their centres, then $|(\alpha+\beta)(\gamma+\delta)|$ is equal to $\_\_\_\_\_\_$ .
Circle
Answer$40$
Solution
Concept. Both centres lie on the normal to the common tangent at the point of contact, $5$ units on either side. Formulas used. (1) Centres $=T\pm5\hat n$, $\hat n=\frac15(4,3)$Step 1 - Check. $(1,2)$ is on $4x+3y=10$ ✓. Step 2 - Use (1). $(1,2)\pm(4,3)$: $C_1=(5,5)$, $C_2=(-3,-1)$.Step 3 - Value. $(\alpha+\beta)(\gamma+\delta)=10\cdot(-4)=-40$; absolute value $40$. Answer: 40