Circle – JEE Advanced Previous Year Questions with Solutions
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Q. Tangents drawn from the point $\mathrm{P}(1,8)$ to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-6 \mathrm{x}-4 \mathrm{y}-11=0$ touch the circle at the points $\mathrm{A}$ and $\mathrm{B}$. The equation of the circumcircle of the triangle $\mathrm{PAB}$ is(A) $\mathrm{x}^{2}+\mathrm{y}^{2}+4 \mathrm{x}-6 \mathrm{y}+19=0$(B) $\mathrm{x}^{2}+\mathrm{y}^{2}-4 \mathrm{x}-10 \mathrm{y}+19=0$(C) $x^{2}+y^{2}-2 x+6 y-29=0$(D) $x^{2}+y^{2}-6 x-4 y+19=0$ [JEE 2009, 3]

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Sol. (B)

Q. The centres of two circles $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ each of unit radius are at a distance of 6 units from eachother. Let $\mathrm{P}$ be the mid point of the line segment joining the centres of $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ and $\mathrm{C}$ be a circle touching circles $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ externally. If a common tangent to $\mathrm{C}_{1}$ and $\mathrm{C}$ passing through $\mathrm{P}$ is also a common tangent to $\mathrm{C}_{2}$ and $\mathrm{C}$, then the radius of the circle $\mathrm{C}$ is [JEE 2009, 4]

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Sol. 8

Q. Two parallel chords of a circle of radius 2 are at a distance $\sqrt{3}+1$ apart. If the chords subtend at the center, angles of $\frac{\pi}{\mathrm{k}}$ and $\frac{2 \pi}{\mathrm{k}},$ where $\mathrm{k}>0,$ then the value of $[\mathrm{k}]$ is[Note : [k] denotes the largest integer less than or equal to k] [JEE 10, 3M]

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Sol. 3

Q. The circle passing through the point (–1,0) and touching the y-axis at (0,2) also passes through the point –(A) $\left(-\frac{3}{2}, 0\right)$(B) $\left(-\frac{5}{2}, 2\right)$(C) $\left(-\frac{3}{2}, \frac{5}{2}\right)$(D) (–4,0) [JEE 2011, 3M, –1M]

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Sol. (D)Family of circle which touches $y$ -axis at $(0,2)$ is$x^{2}+(y-2)^{2}+\lambda x=0$Passing through $(-1,0)$$\Rightarrow 1+4-\lambda=0 \quad \Rightarrow \quad \lambda=5$$\therefore \quad x^{2}+y^{2}+5 x-4 y+4=0$$\quad which satisfy the point (-4,0) Q. The straight line 2 \mathrm{x}-3 \mathrm{y}=1 divides the circular region \mathrm{x}^{2}+\mathrm{y}^{2} \leq 6 into two parts. If S=\left\{\left(2, \frac{3}{4}\right),\left(\frac{5}{2}, \frac{3}{4}\right),\left(\frac{1}{4},-\frac{1}{4}\right),\left(\frac{1}{8}, \frac{1}{4}\right)\right\}, then the number of point(s) in Slying inside the smaller part is [JEE 2011, 4M] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. 2for origin : 2 × 0 – 3 × 0 – 1 = – 1 (–ve) Q. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4 \mathrm{x}-5 \mathrm{y}=20 to the circle \mathrm{x}^{2}+\mathrm{y}^{2}=9 \mathrm{is}-(A) 20\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)-36 \mathrm{x}+45 \mathrm{y}=0(B) 20\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)+36 \mathrm{x}-45 \mathrm{y}=0(C) 36\left(x^{2}+y^{2}\right)-20 x+45 y=0(D) 36\left(x^{2}+y^{2}\right)+20 x-45 y=0 [JEE 2012, 3M, –1M] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (A) Paragraph for Question 7 and 8A tangent PT is drawn to the circle x^{2}+y^{2}=4 at the point P(\sqrt{3}, 1) . A straight line L, perpendicular to PT is a tangent to the circle (x-3)^{2}+y^{2}=1 Q. A common tangent of the two circles is(A) x = 4(B) y = 2(C) x+\sqrt{3} y=4(D) x+2 \sqrt{2} y=6 [JEE 2012, 3M, –1M] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (D) Q. A possible equation of L is(A) x-\sqrt{3} y=1(B) x+\sqrt{3} y=1(C) x-\sqrt{3} y=-1(D) x+\sqrt{3} y=5 [JEE 2012, 3M, –1M] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (A)Equation of tangent at P will be \sqrt{3} x+y=4Slope of line L will be \frac{1}{\sqrt{3}}Let equation of \mathrm{L} be : y=\frac{\mathrm{x}}{\sqrt{3}}+\mathrm{c}$$\Rightarrow \quad x-\sqrt{3} y+\sqrt{3} c=0$Now this $L$ is tangent to $2^{\text {nd }}$ circleSo $\frac{3+\sqrt{3} c}{2}=\pm 1 \quad \Rightarrow \quad c=-\frac{1}{\sqrt{3}}$or $c=-\frac{5}{\sqrt{3}}$using $\quad c=-\frac{1}{\sqrt{3}}$$y=\frac{x}{\sqrt{3}}-\frac{1}{\sqrt{3}} \Rightarrow x-\sqrt{3} y=1 . Hence (\mathrm{A}) Q. Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length 2 \sqrt{7} or y-axis is (are)(A) x^{2}+y^{2}-6 x+8 y+9=0(B) x^{2}+y^{2}-6 x+7 y+9=0(C) \mathrm{x}^{2}+\mathrm{y}^{2}-6 \mathrm{x}-8 \mathrm{y}+9=0(D) x^{2}+y^{2}-6 x-7 y+9=0 [JEE(Advanced) 2013, 3, (–1)] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (A,C) Q. A circle S passes through the point (0,1) and is orthogonal to the circles (x-1)^{2}+y^{2}=16 and x^{2}+y^{2}=1 . Then :-(1) radius of S is 8(B) radius of S is 7(3) centre of S is (–7, 1)(D) centre is S is (–8, 1) [JEE(Advanced)-2014, 3] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (B) Q. Let RS be the diameter of the circle x^{2}+y^{2}=1, where S is the point (1,0). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. then the locus of E passes through the point(s)-(A) \left(\frac{1}{3}, \frac{1}{\sqrt{3}}\right)(B) \left(\frac{1}{4}, \frac{1}{2}\right)(C) \left(\frac{1}{3},-\frac{1}{\sqrt{3}}\right)(D) \left(\frac{1}{4},-\frac{1}{2}\right) (D) \left(\frac{1}{4},-\frac{1}{2}\right) [JEE(Advanced)-2016] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (A,C) Q. For how many values of p, the circle x^{2}+y^{2}+2 x+4 y-p=0 and the coordinate axes have exactly three common points ? [JEE(Advanced)-2017] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (2) Q. Let T be the line passing through the points P(–2, 7) and Q(2, –5). Let \mathrm{F}_{2} be the set of all pairs of circles \left(\mathrm{S}_{1}, \mathrm{S}_{2}\right) such that T is tangents to \mathrm{S}_{1} at P and tangent to \mathrm{S}_{2} at Q, and also such that \mathrm{S}_{1} and \mathrm{S}_{2} touch each other at a point, say, M. Let \mathrm{E}_{1} be the set representing the locus of M as the pair \left(\mathrm{S}_{1}, \mathrm{S}_{2}\right) varies in \mathrm{F}_{1}. Let the set of all straight line segments joining a pair of distinct points of E_{1} and passing through the point R(1, 1) be \mathrm{F}_{2}. Let E_{2} be the set of the mid-points of the line segments in the set \mathrm{F}_{2}. Then, which of the following statement(s) is (are) TRUE ?(A) The point (-2,7) lies in \mathrm{E}_{1}(B) The point \left(\frac{4}{5}, \frac{7}{5}\right) does NOT lie in \mathrm{E}_{2}(C) The point \left(\frac{1}{2}, 1\right) lies in \mathrm{E}_{2}(D) The point \left(0, \frac{3}{2}\right) does NOT lie in \mathrm{E}_{1} [JEE(Advanced)-2018] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (D) Paragraph “X”Let S be the circle in the xy-plane defined by the equation x^{2}+y^{2}=4.(There are two question based on Paragraph “X”, the question given below is one of them) Q. Let \mathrm{E}_{1} \mathrm{E}_{2} and \mathrm{F}_{1} \mathrm{F}_{2} be the chord of S passing through the point \mathrm{P}_{0}(1,1) and parallel to the \mathrm{x}- axis and the \mathrm{y} -axis, respectively. Let \mathrm{G}_{1} \mathrm{G}_{2} be the chord of S passing through \mathrm{P}_{0} and having slop -1 . Let the tangents to \mathrm{S} at \mathrm{E}_{1} and \mathrm{E}_{2} meet at \mathrm{E}_{3}, the tangents of \mathrm{S} at \mathrm{F}_{1} and \mathrm{F}_{2} meet at \mathrm{F}_{3}, and the tangents to \mathrm{S} at \mathrm{G}_{1} and \mathrm{G}_{2} meet at \mathrm{G}_{3} . Then, the points \mathrm{E}_{3}, \mathrm{F}_{3} and \mathrm{G}_{3} lie on the curve(A) x + y = 4(B) (\mathrm{x}-4)^{2}+(\mathrm{y}-4)^{2}=16(C) (x – 4) (y – 4) = 4(D) xy = 4 [JEE(Advanced)-2018] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. (A ) Paragraph “X”Let S be the circle in the xy-plane defined by the equation x^{2}+y^{2}=4(There are two questions based on Paragraph “X”, the question given below is one of them) Q. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve –(A) (\mathrm{x}+\mathrm{y})^{2}=3 \mathrm{xy}(B) \mathrm{x}^{2 / 3}+\mathrm{y}^{2 / 3}=2^{4 / 3}(C) x^{2}+y^{2}=2 x y(D) x^{2}+y^{2}=x^{2} y^{2} [JEE(Advanced)-2018] Download eSaral App for Video Lectures, Complete Revision, Study Material and much more... Sol. 15Tangent at \mathrm{P}(2 \cos \theta, 2 \sin \theta) is \mathrm{xcos} \theta+\mathrm{y} \sin \theta=2$$\mathrm{M}(2 \sec \theta, 0)$ and $\mathrm{N}(0,2 \csc \theta)$Let midpoint be $(\mathrm{h}, \mathrm{k})$$\mathrm{h}=\sec \theta, \quad \mathrm{k}=\csc \theta$$\frac{1}{\mathrm{h}^{2}}+\frac{1}{\mathrm{k}^{2}}=1$$\frac{1}{\mathrm{x}^{2}}+\frac{1}{\mathrm{y}^{2}}=1$

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