JEE Mains · Maths · STD 11 - 10.1 circle and system of circle
Let \(ABCD\) be a square of side of unit length. Let a circle \(C _{1}\) centered at \(A\) with unit radius is drawn. Another circle \(C _{2}\) which touches \(C _{1}\) and the lines \(AD\) and \(AB\) are tangent to it, is also drawn. Let a tangent line from the point \(C\) to the circle \(C _{2}\) meet the side \(AB\) at \(E\). If the length of \(EB\) is \(\alpha+\sqrt{3} \beta,\) where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) is equal to.........
- A \(2\)
- B \(3\)
- C \(1\)
- D \(5\)
Answer & Solution
Correct Answer
(C) \(1\)
Step-by-step Solution
Detailed explanation
Here \(AO + OD =1\) or \((\sqrt{2}+1) r =1\) \(\Rightarrow \quad r =\sqrt{2}-1\) equation of circle \(( x - r )^{2}+( y - r )^{2}= r ^{2}\) Equation of \(CE\) \(y-1=m(x-1)\) \(m x-y+1-M=0\) It is tangent to circle \(\therefore\left|\frac{m r-r+1-m}{\sqrt{m^{2}+1}}\right|=r\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(f\left( x \right) = 5 - \left| {x - 2} \right|\) and \(g\left( x \right) = \left| {x + 1} \right|,x \in R\). If \(f(x)\) attains maximum value at \(\alpha \) and \(g(x)\) attains minimum value at \(\beta \), then \(\mathop {\lim }\limits_{x \to \alpha \beta } \frac{{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)}}{{{x^2} - 6x + 8}}\) is equal toJEE Mains 2019 Hard
- Let \(\mathrm{E}\) be an ellipse whose axes are parallel to the co-ordinates axes, having its center at \((3,-4)\), one focus at \((4,-4)\) and one vertex at \((5,-4) .\) If \(m x-y=4, m\,>\,0\) is a tangent to the ellipse \(\mathrm{E}\), then the value of \(5 \mathrm{~m}^{2}\) is equal to \(.....\)JEE Mains 2021 Hard
- If the circles \({x^2}\, + {y^2}\, - 16x\, - 20y\, + \,164\,\, = \,\,{r^2}\) and \({(x - 4)^2} + {(y - 7)^2} = 36\) intersect at two distinct points, thenJEE Mains 2019 Hard
- Numbers are to be formed between \(1000\) and \(3000\), which are divisible by \(4\), using the digits \(1,2,3,4,5\) and \(6\) without repetition of digits. Then the total number of such numbers is.JEE Mains 2022 Hard
- Statement \(- 1:\) The function \(x^2 (e^x + e^{-x})\) is increasing for all \(x > 0.\) Statement \(-2:\) The functions \(x^2e^x\) and \(x^2e^{-x}\) are increasing for all \(x > 0\) and the sum of two increasing functions in any interval \((a, b)\) is an increasing function in \((a, b).\)JEE Mains 2013 Hard
- If \(\sum_{\mathrm{r}=0}^{10}\left(\frac{10^{\mathrm{r}+1}-1}{10^{\mathrm{r}}}\right) \cdot{ }^{11} \mathrm{C}_{\mathrm{r}+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}\), then \(\alpha\) is equal to :JEE Mains 2025 Hard
More PYQs from JEE Mains
- Let for any three distinct consecutive terms \(a, b, c\) of an \(A.P,\) the lines \(a x+b y+c=0\) be concurrent at the point \(\mathrm{P}\) and \(\mathrm{Q}(\alpha, \beta)\) be a point such that the system of equations \( x+y+z=6, \) \( 2 x+5 y+\alpha z=\beta\) and \(x+2 y+3 z=4\), has infinitely many solutions. Then \((P Q)^2\) is equal to ...........JEE Mains 2024 Hard
- If the equation of the plane passing through the point \((1,1,2)\) and perpendicular to the line \(x-3 y+2 z-1=04 x-y+z\) is \(Ax + By + Cz =1\), then \(140( C - B + A )\) is equal to \(.........\).JEE Mains 2023 Hard
- If the sum of series \(\frac{1}{1 \cdot(1+d)}+\frac{1}{(1+d)(1+2 d)}+\ldots \ldots+\frac{1}{(1+9 d)(1+10 d)}\) is equal to \(5\) , then \(50 \mathrm{~d}\) is equal to :JEE Mains 2024 Hard
- If the function \(f\left( x \right) = \left\{ \begin{array}{l}
a\,\left| {\pi - x} \right|\, + 1,\,\,x \le 5\,\\
b\,\,\left| {\pi - x} \right|\, + 3,\,\,x > 5\,\,
\end{array} \right.\) is continuous at \(x = 5\), then the value of \(a -b\) isJEE Mains 2019 Hard - The area of the region enclosed between the parabolas \(y ^{2}=2 x -1\) and \(y ^{2}=4 x -3\) isJEE Mains 2022 Medium
- If \(y = {\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}}\) , then \(\left( {{x^2} - 1} \right)\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}}\) is equal toJEE Mains 2017 Hard