MHT CET · Maths · Circle

The co-ordinates of the mid-point of the chord cut off by theline \(2 x-5 y+18=0\)
by the circle \(x^{2}+y^{2}-6 x+2 y-54=0\) are

A \((1,4)\)
B \((2,4)\)
C \((4,1)\)
D \((1,1)\)

Verified Solution

Answer & Solution

Correct Answer

(A) \((1,4)\)

Step-by-step Solution

Detailed explanation

Let \(A B\) be the chord
Let the mid point of chord be \(\mathrm{M}(\mathrm{h}, \mathrm{k})\)
Here centre is \(\mathrm{O}=(3-1)\)
Here \(O M \perp A B\)
\(\therefore(\) slope of \(\mathrm{OM})(\) slope of \(\mathrm{AB})=-1\)
\(\left(\frac{\mathrm{k}+1}{\mathrm{~h}-3}\right)\left(\frac{2}{5}\right)=-1\)
\(\therefore 5 \mathrm{~h}+2 \mathrm{k}=13\)
Point \(M(h, k)\) lies on the \(2 x-5 y+18=0\)...(1)
\(\therefore 2 \mathrm{~h}-5 \mathrm{k}=-18\)...(2)
Solving equation (1) \& (2) we get \(\mathrm{h}=1, \mathrm{k}=4 \Rightarrow(1,4)\) is required point.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The co-ordinates of the mid-point of the chord cut off by theline \(2 x-5 y+18=0\)by the circle \(x^{2}+y^{2}-6 x+2 y-54=0\) are

Step-by-step Solution

The co-ordinates of the mid-point of the chord cut off by theline \(2 x-5 y+18=0\)
by the circle \(x^{2}+y^{2}-6 x+2 y-54=0\) are