Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Consider ellipse<img alt="\mathrm{E}_{\mathrm{k}}: \mathrm{kx}^2+\mathrm{k}^2 \mathrm{y}^2=1, \mathrm{k}=1,2, \ldots, 20" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BE%7D_%7B%5

Consider ellipse\mathrm{E}_{\mathrm{k}}: \mathrm{kx}^2+\mathrm{k}^2 \mathrm{y}^2=1, \mathrm{k}=1,2, \ldots, 20. Let Ck be the circle which touches the four chords joining
the end points (one on minor axis and another on major axis) of the ellipse Ek.If rk is the radius of the circle Ck then the value of\sum_{k=1}^{20} \frac{1}{r_k^2} \text { is }

Option: 1

3320


Option: 2

3210


Option: 3

3080


Option: 4

2870


Answers (1)

best_answer

\mathrm{Kx}^2+\mathrm{K}^2 \mathrm{y}^2=1 \frac{x^2}{1 / K}+\frac{y^2}{1 / K^2}=1 Now

Equation of $$ \begin{aligned} & \mathrm{A}_1 \mathrm{~B}_2 ; \frac{\mathrm{x}}{1 / \sqrt{\mathrm{K}}}+\frac{\mathrm{y}}{1 / \mathrm{K}}=1 \Rightarrow \sqrt{\mathrm{K} x}+\mathrm{Ky}=1 \\ & \mathrm{r}_{\mathrm{K}}=\perp \mathrm{r} \text { distance of }(0,0) \text { from line } \mathrm{A}_1 \mathrm{~B}_1 \\ & \mathrm{r}_{\mathrm{K}}=\left|\frac{(0+0-1)}{\sqrt{\mathrm{K}+\mathrm{K}^2}}\right|=\frac{1}{\sqrt{\mathrm{K}+\mathrm{K}^2}} \\ & \frac{1}{\mathrm{r}_{\mathrm{K}}^2}=\mathrm{K}+\mathrm{K}^2 \Rightarrow \sum_{\mathrm{k}=1}^{20} \frac{1}{\mathrm{r}_{\mathrm{K}}^2}=\sum_{\mathrm{K}=1}^{20}\left(\mathrm{~K}+\mathrm{K}^2\right) \\ & =\sum_{\mathrm{K}=1}^{20} \mathrm{~K}+\sum_{\mathrm{K}=1}^{20} \mathrm{~K}^2 \\ & =\frac{20 \times 21}{2}+\frac{20.21 .41}{6} \\ & =210+10 \times 7 \times 41 \\ & =210+2870 \\ & =3080 \end{aligned}

Posted by

Irshad Anwar

View full answer

JEE Main high-scoring chapters and topics

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

Similar Questions

Trending Articles/News

anam-khan

Calculate JEE Main 2025 rank using predictor tool for admission to top IITs, NITs, IIITs
anam-khan

NTA drops 1 out of 9 flagged errors from JEE Mains final answer key 2025
anam-khan

Consumer protection watchdog warns coaching centres against misleading ads ahead of IIT-JEE Main results

Latest Question