Question

The length of the chord of the circle ${{x}^{2}}+{{y}^{2}}+4x-7y+12=0$ along the y-axis is
(A) $1$
(B) $2$
(C) $\dfrac{1}{2}$
(D) None of these

Answer 1

Hint: For answering this question we will put $x=0$ in the given equation of the circle ${{x}^{2}}+{{y}^{2}}+4x-7y+12=0$ and obtain the equation of the chord along the y-axis and find the two endpoints of it and derive the distance between them which will be equal to the length of the chord.

Complete step-by-step solution:
We have the equation of the circle from the question as ${{x}^{2}}+{{y}^{2}}+4x-7y+12=0$.
The equation of the chord of this circle along the y-axis is given by substituting $x=0$ in the equation.
After substituting we will have ${{y}^{2}}-7y+12=0$ .
This is a quadratic equation which can be simply written as ${{y}^{2}}-3y-4y+12=0\Rightarrow \left( y-4 \right)\left( y-3 \right)=0$ .
Hence, 3 and 4 are the factors of this equation. So we can say that $\left( 0,3 \right)$ and $\left( 0,4 \right)$ are the endpoints of the chord.
Now the length of the chord can be derived by finding the distance between the two endpoints.
From the basic concept we know that for any two points lying on the y-axis the distance between them is given as $\left| {{y}_{1}}-{{y}_{2}} \right|$.
Hence the length of the chord of the circle ${{x}^{2}}+{{y}^{2}}+4x-7y+12=0$ along the y-axis is given as $1$.

Hence, option A is correct.

Note: While answering this type of question we should take care while obtaining the endpoints of the chord by deriving the roots of the equation of the chord if in case we made a mistake and obtained the roots as follows ${{y}^{2}}-7y+12=0\Rightarrow \left( y-5 \right)\left( y-3 \right)=0$ so we will have the answer as the option B which is wrong.