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Circle

Section: Mathematics

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Question : 5 of 89

Marks: +1, -0

If the circles

x^2+y^2+5k x+2y+k=0

2x^2+2y^2+2k x+3y-1=0, k \in \mathbb{R}

intersect at points P and Q , then the line

4x+5y-k=0

passes through P and Q for

[AP EAPCET 23 May 2025 S2]

exactly one value of $k$
exactly two values of k
no value of $k$
infinitely many values of $k$

Solution:

Given equation of circle

x^2+y^2+5k x+2y+k=0

and

2x^2+2y^2+2k x+3y-1=0

Equation of common chord

8k x+y+2k+1=0

Since, this chord is coincide with

\;4x+5y-k=0

\;\;\frac{8k}{4}=\;\frac{1}{5}=\;\frac{2k+1}{-k}

Solving

\;\frac{8k}{4}=\;\frac{1}{5}

we get

k=\;\frac{1}{10}

Solving

8\;\frac{k}{4}=\;\frac{2k+1}{-k} \Rightarrow 2k^2+2k+1=0

No solution

∴ No values of

k

possible

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