Go To Your Study Dashboard

Get Answers to all your Questions

header-bg

Let a circle $C_1$ be obtained on rolling the circle $x^{2}+y^{2} -4x-6y+11=0$ upwards 4 units on the tangent T to it at the point (3,2). Let $C_2$ be the image of $C_1$ in T. Let A and B be the centers of circles $C_1$ and $C_2$ respectively, and M and N be respectively the feet of perpendiculars drawn from A and B on the x-axis. Then the area of the trapezium AMNB is :
Option: 1
$4(1+\sqrt2)$

Option: 2
$3+2\sqrt2$

Option: 3
$2(1+\sqrt2)$

Option: 4
$2(2+\sqrt2)$

Answers (1)

best_answer

(x', y') point lies on line y = x – 1 have distance 4 unit from (3, 2).

$\begin{aligned} & x^{\prime}=\frac{4}{\sqrt{2}}+3=2 \sqrt{2}+3 \\ & y^{\prime}=\frac{4}{\sqrt{2}}+2=2 \sqrt{2}+2 \end{aligned}$

Slope of line AB is –1. $\text { i.e. }=\tan \theta=-1 \text { then } \sin \theta=\frac{1}{\sqrt{2}}, \cos \theta=-\frac{1}{\sqrt{2}}$

for point A and B

$\begin{aligned} & x= \pm \sqrt{2}\left(\frac{-1}{\sqrt{2}}\right)+(2 \sqrt{2}+3) \\ & y= \pm \sqrt{2}\left(\frac{1}{\sqrt{2}}\right)+(2 \sqrt{2}+2) \end{aligned}$

for point A we take +ve sign

$(x_2 ,\, y_2)=(2\sqrt2+2,\, \, 2\sqrt2+3)$

for point B we take –ve sign

$(x_1 ,\, y_1)=(2\sqrt2+4,\, \, 2\sqrt2+1)$

MN = $\left|x_2-x_1\right|=2$

AM+BN = $2\sqrt2+3+2\sqrt2+1=4+4\sqrt2$

area of trapezium $\begin{gathered} =\frac{1}{2} \times 2 \times(4+4 \sqrt{2}) \\ \end{gathered}$

$\begin{gathered} \quad=4(1+\sqrt{2}) \end{gathered}$

Posted by

rishi.raj

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

50 mL of 0.2 M ammonia solution is treated with 25 mL of 0.2 M HCl. If pK_b of ammonia solution is 4.75, the pH of the mixture will be :
Option: 1 3.75
Option: 2 4.75

Trending Articles/News

anam-khan

JEE Main Passing Marks 2025: Cut-off rises by 3 marks for General; category-wise qualifying scores

anam-khan

JEE Main 2025 Live: NTA session 1 exam city slip soon at jeemain.nta.nic.in; schedule, cut-offs

anam-khan

JEE Main 2025 Live: NTA JEE exam city slip at jeemain.nta.nic.in soon; exam dates, cut-offs

Latest Question