NIMCET 2017 — Mathematics PYQ

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, $\varphi (x,y)=x^2+3xy+2y^2-x-4y-6=0$
Taking partial derivative w.r.t. x, we get
$$\frac{d\varphi }{dx}=2x+3y-1=0\cdot \cdot \cdot (1)$$
Taking partial derivative w.r.t. x, we get
$$<br>\frac{d\varphi }{dy}=3x+4y-4=0\cdot \cdot \cdot (2)$$
Multiplying (1) by 3 and (2) by 2, we get
$$6x+9y-3=0$$... (3) and $$6x+8y-8=0$$... (4)
Now, subtracting (4) from (3), we get
$$<br>y+5=0\Rightarrow y=-5$$
From (1),
$$2x+3(-5)-1=0$$
$$<br>\Rightarrow x=8$$
So, the point of intersection is (8, -5)
Hence, option (3) is correct.