The line $(2+k)x+(1+k)y = 5+7k$ passes through the fixed point for different values of k. If 'd' is the distance of a fixed point from the origin, then $d^2 = \ldots$

Question

If $ x = a(\cos \theta + \theta \sin \theta), \, y = a(\sin \theta - \theta \cos \theta) $, then $ \frac{d^2 y}{dx^2} $ equals

Answer 1

Step 1: Pick two values of k
For $k=0$: $2x+y = 5$. For $k=-1$: $x = -2$.

Step 2: Intersect
Put $x=-2$ in the first: $y = 9$. The point is $(-2,9)$ and it must lie on all lines of the family.

Step 3: Verify and compute
Check with the general form gives $5+7k$ on both sides. Then $d^2 = 4+81 = 85$. Option (D).

Final Answer:
85. \[ \boxed{\text{(D)}\ 85} \]

The line \((2+k)x+(1+k)y = 5+7k\) passes through the fixed point for different values of k. If 'd' is the distance of a fixed point from the origin, then \(d^2 = \ldots\)

Show Hint

The Correct Option is D

Solution and Explanation

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