Correct Answer - Option 3 : (8, -5)
Concept:
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.
Example: partial differentiate w.r.t. x then y will be constant and vice versa.(for function containing x and y variables).
Calculation:
Let, ϕ (x, y) = x2 + 3xy + 2y2 - x - 4y - 6 = 0
Taking partial derivative w.r.t. x, we get
\(\rm \frac{dϕ }{dx}=2x+3y-1=0\cdots (1)\)
Taking partial derivative w.r.t. x, we get
\(\rm \frac{dϕ }{dy}=3x+4y-4=0\cdots (2)\)
Multiplying (1) by 3 and (2) by 2, we get
\(\rm 6x+9y-3=0\cdots (3)\) and \(\rm 6x+8y-8=0\cdots (4)\)
Now , subtracting (4) from (3), we get
y + 5 = 0 ⇒ y = -5
From (1), 2x + 3(-5) -1 = 0
⇒ x = 8
So, the point of intersection is (8, -5)
Hence, option (3) is correct.