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  • Number of the lines, among the lines <img alt="\mathrm{2 x+y+4+k(x-2 y-3)=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B2%20x&plus;y&plus;4&plus;k%28x-2%20y-3%29%

Number of the lines, among the lines \mathrm{2 x+y+4+k(x-2 y-3)=0}; whose minimum distance is \mathrm{\sqrt{10}} from the point \mathrm{(2,-3)} is

Option: 1

1


Option: 2

2


Option: 3

0


Option: 4

infinite


Answers (1)

best_answer

We have \mathrm{\left|\frac{(2+k) 2+(1-2 k)(-3)+4-3 k}{\sqrt{(2+k)^{2}+(1-2 k)^{2}}}\right|=\sqrt{10}}

On simplifying this quadratic will yield equal root \mathrm{k=1}.

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