JEE Mains · Maths · STD 11 - 10.1 circle and system of circle
Let \(y=x\) be the equation of a chord of the circle \(C_{1}\) (in the closed half-plane \(x\ge0)\) of diameter 10 passing through the origin. Let \(C_{2}\) be another circle described on the given chord as its diameter. If the equation of the chord of the circle \(C_{2}\), which passes through the point (2, 3) and is farthest from the center of \(C_{2}\), is \(x+ay+b=0,\) then \(a-b\) is equal to:
- A 10
- B -6
- C -2
- D 6
Answer & Solution
Correct Answer
(C) -2
Step-by-step Solution
Detailed explanation
Equation of circle \(C_{2}\) is \(x^{2}+y^{2}-5x-5y=0\) its centre is \((\frac{5}{2},\frac{5}{2})\) \(m_{AB}=-1\) \(\therefore\) Slope of required chord \(=1\) \(\therefore\) equation of required chord is \(x - y + 1 =0\) \(\therefore a =-1, b=2\) \(\therefore a - b =-2\)
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 \( \vec{c} \) and \( \vec{d} \) be vectors such that \( |\vec{c}+\vec{d}|=\sqrt{29} \) and \( \vec{c}\times(2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times\vec{d} \). If \( \lambda_1, \lambda_2 \) (\(\lambda_1 \)>\(\lambda_2 \)) are the possible values of \( (\vec{c}+\vec{d}).(-7\hat{i}+2\hat{j}+3\hat{k}) \),
then the equation
\( K^{2}x^{2}+(K^{2}-5K+\lambda_{1})xy+(3K+\frac{\lambda_{2}}{2})y^{2}-8x+12y+\lambda_{2}=0 \)
represents a circle, for k equal to:JEE Mains 2026 Easy - Let \(\vec{a}=\vec{i}-\alpha \vec{j}+\beta \hat{k}, \vec{b}=3 \hat{i}+\beta \hat{j}-\alpha \hat{k}\) and \(\vec{c}=-\alpha \hat{i}-2 \hat{j}+\hat{k}\), where \(\alpha\) and \(\beta\) are integers. If \(\vec{a} \cdot \vec{b}=-1\) and \(\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=10\), then \((\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{c}}\) is equal to \(.....\)JEE Mains 2021 Hard
- Let in a \(\triangle A B C\), the length of the side \(A C\) be 6 , the vertex \(B\) be \((1,2,3)\) and the vertices \(A, C\) lie on the line \(\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\). Then the area (in sq. units) of \(\triangle \mathrm{ABC}\) is:JEE Mains 2025 Medium
- The function \(f(x)=x^{3}-6 x^{2}+a x+b\) is such that \(f(2)=f(4)=0\). Consider two statements. \((S_1)\) there exists \(\mathrm{x}_{1}, \mathrm{x}_{2} \in(2,4), \mathrm{x}_{1}<\mathrm{x}_{2}\), such that \(f^{\prime}\left(x_{1}\right)=-1\) and \(f^{\prime}\left(x_{2}\right)=0\) \((S_2)\) there exists \(\mathrm{x}_{3}, \mathrm{x}_{4} \in(2,4), \mathrm{x}_{3}<\mathrm{x}_{4}\), such that \(f\) is decreasing in \(\left(2, x_{4}\right)\), increasing in \(\left(x_{4}, 4\right)\) and \(2 f^{\prime}\left(x_{3}\right)=\sqrt{3} f\left(x_{4}\right)\). ThenJEE Mains 2021 Hard
- Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :JEE Mains 2025 Medium
- A tangent to the hyperbola \(\frac{{{x^2}}}{4} - \frac{{{y^2}}}{2} = 1\) meets \(x-\) axis at \(P\) and \(y-\) axis at \(Q\). Lines \(PR\) and \(QR\) are drawn such that \(OPRQ\) is a rectangle (where \(O\) is the origin). Then \(R\) lies onJEE Mains 2013 Hard
More PYQs from JEE Mains
- Let \(\mathrm{n}\) denote the number of solutions of the equation \(z^{2}+3 \bar{z}=0\), where \(\mathrm{z}\) is a complex number. Then the value of \(\sum_{k=0}^{\infty} \frac{1}{n^{k}}\) is equal to:JEE Mains 2021 Hard
- Let \(A=\{1,2,3\}\). The number of relations on \(A\), containing \((1,2)\) and \((2,3)\), which are reflexive and transitive but not symmetric, is ______ -JEE Mains 2025 Easy
- Let \(\theta=\frac{\pi}{5}\) and \(A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right] \cdot\) If \(B=A + A ^{4},\) then \(\operatorname{det}( B )\)JEE Mains 2020 Hard
- If \(A\) and \(B\) are two non-zero \(n \times n\) matrics such that \(A ^2+ B = A ^2 B\), thenJEE Mains 2023 Hard
- Let the median and the mean deviation about the median of \(7\) observation \(170,125,230,190,210\), \(a, b\) be 1\(70\) and \(\frac{205}{7}\) respectively. Then the mean deviation about the mean of these \(7\) observations is :JEE Mains 2024 Hard
- Let the line \(\mathrm{L}\) intersect the lines \(\mathrm{x}-2=-\mathrm{y}=\mathrm{z}-1,2(\mathrm{x}+1)=2(\mathrm{y}-1)=\mathrm{z}+1\) and be parallel to the line \(\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}\). Then which of the following points lies on \(\mathrm{L}\) ?JEE Mains 2024 Hard