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Intercepts Made by Circle on the Axis - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Different Form of the Equation of the Circle is considered one the most difficult concept.

  • 34 Questions around this concept.

Solve by difficulty

QuestionEASY

 The centres of those circles which touch the circle, x2+y2−8x−8y−4=0, externally and also touch the x-axis, lie on :

 

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The circle passing through (1,-2) and touching the axis of x at (3,0) also passes through the point :

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 Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centred at (0, y), passing through origin and touching the circle C externally, then the radius of T is equal to :

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A circle touches the lines \mathrm{y=\frac{x}{\sqrt{3}}, y=\sqrt{3} x} the centre of this circle lies in the first quadrant then one possible equation of this circle is

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Equation of circle touching the lines \mathrm{|x-2|+|y-3|=4}  will be

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Four distinct points (2 K, 3 K),(1,0),(0,1)$ and $(0,0) lie on a circle when:

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Concepts Covered - 2

Intercepts Made by Circle on the Axis

Intercepts Made by Circle on the Axis 
If the equation of Circle is \mathrm{x^2+y^2+2gx+2fy+c=0}, then 

Length of x-intercept :\mathrm{2\sqrt{g^2-c}}

Length of y-intercept : \mathrm{2\sqrt{f^2-c}}

 

Proof:

\begin{array}{l}{\text { from the figure }} \\ {\text { length of intercepts on } X-\text { axis and } Y-\text { axis are }|A B| \text { and }|C D|} \\ {|A B|=\left|x_{2}-x_{1}\right|,|C D|=\left|y_{2}-y_{1}\right|} \\ \\ {Put\,\,y=0, \text {to get points A and B, where circle intersects the } X-\text { axis }} \\ {\Rightarrow x^{2}+2 g x+c=0} \\ {\text {Since, circle intersects } X-\text { axis at two points } A\left(x_{1}, 0\right) \text { and } B\left(x_{2}, 0\right)} \\ {\text {so x1 and x2 are roots of the above equation, and hence, } x_{1}+x_{2}=-2 g x, x_{1} x_{2}=c} \\ {|A B|=\left|x_{2}-x_{1}\right|=\sqrt{\left(x_{2}+x_{1}\right)^{2}-4 x_{1} x_{2}}} =2\sqrt{g^2-c}\\\\ {\text {Similarly, }} \\ {|C D|=2 \sqrt{f^{2}-c}}\end{array}

Different Form of the Equation of the Circle

Different Form of a Circle

When the circle touches X-axis

(a,b) be the centre of the circle, then radius = |b|

\\\therefore \text{equation of circle becomes} \\\Rightarrow\;\mathrm{(x-a)^{2}+(y-b)^{2}=b^{2}} \\\mathbf{\Rightarrow\; x^{2}+y^{2}-2 a x-2 b y+a^{2}=0}

 

When the circle touches Y-axis

(a,b) be the centre of the circle, then radius = |a|

\\\therefore \text{equation of circle becomes} \\\Rightarrow\;\mathrm{(x-a)^{2}+(y-b)^{2}=a^{2}} \\\mathbf{\Rightarrow\; x^{2}+y^{2}-2 a x-2 b y+b^{2}=0}

 

When the circle touches both the axes:

(a,a) be the centre of the circle, then radius = |a|

\\\therefore \text{equation of circle becomes} \\\Rightarrow\;\mathrm{(x-a)^{2}+(y-a)^{2}=a^{2}} \\\mathbf{\Rightarrow\; x^{2}+y^{2}-2 a x-2 a y+a^{2}=0}

Note:

In this case, the centre can also be (a, -a) and radius |a|.

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Intercepts Made by Circle on the Axis
Different Form of the Equation of the Circle

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Transformations of Axes

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Reference Books

Different Form of the Equation of the Circle

Mathematics for Joint Entrance Examination JEE (Advanced) : Coordinate Geometry

Page No. : 4.4

Line : 37

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