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CAT Position of a Point with Respect to a Circle - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Syllabus

Quick Facts

5 Questions around this concept.

Solve by difficulty

EASY

Which of the following circle has (5,0) on it:

$x^2+y^2=5$

$x^2+y^2+2gx+4y-16=0$

$x^2+y^2-5x-5y=0$

$x^2+y^2=25$

View Solution

Which of the following points lies outside the circle $x^{2}+y^{2}-4x+2y-10=0$ ?

(2,6)

(2,-3)

(4,6)

Both (a) and (c)

View Solution

Concepts Covered - 1

Position of a Point with Respect to a Circle

If we have given equation of circle, let say S : x² + y² + 2gx +2fy + c = 0 and any point P(x₁, y₁) in same plane, then we can tell weather this point P lies inside, outside or on the circle.

How?

Let's consider the figure

Point P (x₁, y₁) lies outside, on or inside the circle, S accordingly as CP is greater than, equal to or less than the radius, respectively.

Distance between the point C(-g,-f) and P(x₁, y₁) is $\mathrm{\sqrt{\left(x_{1}-(-g)\right)^{2}+\left(y_{1}-(-f)\right)^{2}}}$ and radius of circle, S is $\mathrm{\sqrt{g^2+f^2-c}}$ .

So,

$\begin{aligned} \mathrm{\sqrt{\left(x_{1}-(-g)\right)^{2}+\left(y_{1}-(-f)\right)^{2}}}&>,=,<\mathrm{\sqrt{g^{2}+f^{2}-c}}\\ \mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}+2 \mathrm{gx}_{1}+2 \mathrm{fy}_{1}+\mathrm{c}&>,=,<0\\ \mathrm{S}_{1}&>,=,<0 \end{aligned}$

(a) Point P lies outside the circle if S₁ > 0.
(b) Point P lies on the circle (on the circumference) if S₁ = 0
(c) Point P lies inside the circle if S₁ < 0

$\mathrm{Here,\;\;S_1=x^2_1+y^2_1+2gx_1+2fy_1+c}$

Also, If we have given an equation of circle and and any point in a same plane then, we can tell, how many tangent(s) can be drawn from the given point to the circle.

(a) Point P lies outside the circle, so S₁ > 0, hence two tangent can be drawn.
(b) Point P lies on the circle (on the circumference), so S₁ = 0, hence one tangent is possible
(c) Point P lies inside the circle, so S₁ < 0, hence no tangent is possible.