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Question: Without changing the direction of the coordinate axes, origin is transferred to (h,k) so that the li...

Without changing the direction of the coordinate axes, origin is transferred to (h,k) so that the linear (one degree) terms in the equation x2+y2−4x+6y−7=0{{x}^{2}}+{{y}^{2}}-4x+6y-7=0 are eliminated. Then the point (h,k) is eliminated. Then the point (h,k) is
a)(3,2)
b)(-3,2)
c)(2,-3)
d)None of these

Explanation

Solution

Hint: In this question, we need to find the change in coordinates such that the linear terms get eliminated. Therefore, we should understand the change in the equation of curve under coordinate transformation and then put the conditions to obtain the equations by solving which we can get the required answer.

Complete step-by-step answer:
We know that the when the origin is transferred to the point (h,k), the new x coordinate will be the distance from the new y-axis (which is obtained by the condition that the points on it have the x-component of the new coordinate system zero) and the new y coordinate will be given by the distance from the distance from the new x-axis.
Thus, if the new coordinates are given by X and Y, then the point x and y can be written in the new coordinate system as
x=h+Xx=h+X and y=k+Y.................(1.1)y=k+Y.................(1.1)
Therefore, we can replace x and y in the equation given in the question to obtain
x2+y2−4x+6y−7=0⇒(h+X)2+(k+Y)2−4(h+X)+6(k+Y)−7=0 ⇒h2+2hX+X2+k2+2kY+Y2−4h−4X+6k+6Y−7=0 ⇒X2+Y2+(2h−4)X+(2k+6)Y+h2+y2−7=0 \begin{aligned} & {{x}^{2}}+{{y}^{2}}-4x+6y-7=0\Rightarrow {{\left( h+X \right)}^{2}}+{{\left( k+Y \right)}^{2}}-4\left( h+X \right)+6\left( k+Y \right)-7=0 \\\ & \Rightarrow {{h}^{2}}+2hX+{{X}^{2}}+{{k}^{2}}+2kY+{{Y}^{2}}-4h-4X+6k+6Y-7=0 \\\ & \Rightarrow {{X}^{2}}+{{Y}^{2}}+\left( 2h-4 \right)X+\left( 2k+6 \right)Y+{{h}^{2}}+{{y}^{2}}-7=0 \\\ \end{aligned}
As, it is given that the coefficients of the linear terms should be zero, we should have
2h−4=0⇒h=22h-4=0\Rightarrow h=2
And
2k+6=0⇒k=−32k+6=0\Rightarrow k=-3
Which matches option (c). Therefore, the answer to this question should be option (c).

Note: We should note that the old coordinates are obtained by adding the new X and Y coordinates with the corresponding coordinates of the new origin. We should not subtract the coordinates of the new system with that of the origin as this would result in a shifting of the coordinates in the opposite direction.