JEE Mains · Maths · STD 11 - 10.1 circle and system of circle
If the variable line \(3 x+4 y=\alpha\) lies between the two circles \((x-1)^{2}+(y-1)^{2}=1\) and \((x-9)^{2}+(y-1)^{2}=4\) without intercepting a chord on either circle, then the sum of all the integral values of \(\alpha\) is .... .
- A \(164\)
- B \(166\)
- C \(165\)
- D \(199\)
Answer & Solution
Correct Answer
(C) \(165\)
Step-by-step Solution
Detailed explanation
Both centres should lie on either side of the line as well as line can be tangent to circle. \((3+4-\alpha) \cdot(27+4-\alpha)\,<\,0\) \((7-\alpha) \cdot(31-\alpha)<0 \Rightarrow \alpha \in(7,31) \quad \ldots(1)\) \(d_{1}=\text { distance of }(1,1) \text { from line }\)…
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
- The product of the last two digits of \((1919)^{1919}\) is ___________.JEE Mains 2025 Easy
- If the area of the triangle formed by the positive \(x-\)axis, the normal and the tangent to the circle \((x-2)^{2}+(y-3)^{2}=25\) at the point \((5,7)\) is \(A\) then \(24 A\) is equal to ...... .JEE Mains 2021 Hard
- Let \(f\) be a differentiable function such that \(2(x+2)^2 f(x)-3(x+2)^2=10 \int_0^x(t+2) f(t) d t, x \geq 0\). Then \(f(2)\) is equal to ______.JEE Mains 2025 Hard
- Let \(y = y(x)\) be the solution of the differential equation \((x^2 - x\sqrt{x^2 - 1})dy + (y(x - \sqrt{x^2 - 1}) - x)dx = 0\), \(x \geq 1\). If \(y(1) = 1\), then the greatest integer less than \(y(\sqrt{5})\) is _______.JEE Mains 2026 Hard
- Let the triangle PQR be the image of the triangle with vertices \((1,3),(3,1)\) and \((2,4)\) in the line \(x+2 y=2\). If the centroid of \(\triangle \mathrm{PQR}\) is the point \((\alpha, \beta)\), then \(15(\alpha-\beta)\) is equal to :JEE Mains 2025 Hard
- Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function defined by \(f(x)=\|x+2|-2| x\|\). If \(m\) is the number of points of local minima and \(n\) is the number of points of local maxima of \(f\), then \(m+n\) isJEE Mains 2025 Easy
More PYQs from JEE Mains
- If the area of the region \(\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^2\right\}\right\}\) is \(A\), then \(12 A\) is equal toJEE Mains 2024 Medium
- The probability distribution of random variable \(\mathrm{X}\) is given by:
Let \(\mathrm{p}=\mathrm{P}(1\,<\mathrm{X}\,<\,4 \mid \mathrm{X}\,<\,3)\). If \(5 \mathrm{p}=\lambda \mathrm{K}\), then \(\lambda\) equal to .... .\(X\) \(1\) \(2\) \(3\) \(4\) \(5\) \(P(X)\) \(K\) \(2K\) \(2K\) \(3K\) \(K\) JEE Mains 2021 Medium - Let \(\vec{a}=\hat{i}+\alpha \hat{j}+3 \hat{k}\) and \(\vec{b}=3 \hat{i}-\alpha \hat{j}+\hat{k} \cdot\) If the area of the parallelogram whose adjacent sides are represented by the vectors \(\vec{a}\) and \(\vec{b}\) is \(8 \sqrt{3}\) square units, then \(\overrightarrow{ a } \cdot \overrightarrow{ b }\) is equal to ....... .JEE Mains 2021 Medium
- Let \(z = 1 + ai\) be a complex number, \(a > 0\), such that \(z^3\) is areal number. Then the sum \(1 + z + z^2 + .... + z^{11}\) is equal toJEE Mains 2016 Hard
- If \(\mathrm{a}_{\mathrm{r}}=\cos \frac{2 \mathrm{r} \pi}{9}+i \sin \frac{2 \mathrm{r} \pi}{9}, \mathrm{r}=1,2,3, \ldots, i=\sqrt{-1}\) then the determinant \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|\) is equal to :JEE Mains 2021 Medium
- If the arithmetic mean of two numbers \(a\) and \(b, a>b>0\), is five times their geometric mean, then \(\frac{{a + b}}{{a - b}}\) is equal toJEE Mains 2017 Hard