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Circle

Section: Mathematics

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Question : 21 of 89

Marks: +1, -0

A circle touches the line

2 x+y-10=0

(3,4)

and passes through the point

(1,-2)

. Then a point that lies on the circle is

[AP EAPCET 21 May 2025 S2]

$(5,4)$
$(4,5)$
$(-5,4)$
$(4,-5)$

Solution:

Given, line

2 x+y-10=0

touches the circle at the point

(3,4)

.

\because

Perpendicular to this line passes through centre of circle.

Equation of line perpendicular to

2 x+y-10=0

is

\; x-2 y+\lambda=0

\Rightarrow 3-8+\lambda=0 \Rightarrow \lambda=5

\therefore \; x-2 y+5=0

Centre of the circle be

C(2k-5,k)

Now radius

r^2=CP^2=CQ^2

r^2=(2k-8)^2+(k-4)^2=(2k-6)^2+(k+2)^2

On solving this, we get

k=2

\because

Centre

(-1,2)

and radius

=\sqrt{20}

Equation of circle be

(x+1)^2+(y-2)^2=20

Clearly

(-5,4)

lies on the circle.

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