questionr
Early access
Exams
Joint Entrance Examination
Mathematics
Circle
Question 41
Question 41 of 219
Medium
MCQ
Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and...
A
r
=
1
r=1
r
=
1
B
2
r
2
−
4
r
+
1
=
0
2{r}^{2}-4r+1=0
2
r
2
−
4
r
+
1
=
0
C
2
r
2
−
8
r
+
7
=
0
2{r}^{2}-8r+7=0
2
r
2
−
8
r
+
7
=
0
D
r
2
−
8
r
+
8
=
0
{r}^{2}-8r+8=0
r
2
−
8
r
+
8
=
0
Check answer
Press A-D to select · Enter to check
Solutions · 1
Post your solution
Official solution
\[C F=4 \sqrt{2}-2 \sqrt{2}=2 \sqrt{2}=r+r \sqrt{2}\] \[
⇒
(
2
−
r
)
2
=
r
⇒
2
=
(
r
2
−
r
)
⇒
2
=
r
2
(
2
−
r
)
2
⇒
2
(
r
2
−
4
r
+
4
)
=
r
2
⇒
r
2
−
8
r
+
8
=
0
\begin{aligned} & \Rightarrow \quad(2-r) \sqrt{2}=r \\ & \Rightarrow \quad \sqrt{2}=\left(\frac{r}{2-r}\right) \Rightarrow 2=\frac{r^2}{(2-r)^2} \\ & \Rightarrow 2\left(r^2-4 r+4\right)=r^2 \\ & \Rightarrow r^2-8 r+8=0 \end{aligned}
⇒
(
2
−
r
)
2
=
r
⇒
2
=
(
2
−
r
r
)
⇒
2
=
(
2
−
r
)
2
r
2
⇒
2
(
r
2
−
4
r
+
4
)
=
r
2
⇒
r
2
−
8
r
+
8
=
0
\]
Answer the question to unlock
Previous
Next