Question 20.11.5: Find the point to which the origin be shifted after a transl......
Find the point to which the origin be shifted after a translation, so that the equation x^{2}+y^{2}-4 x-8 y+3=0 will have no first degree terms.
Step-by-Step
Let the origin O be shifted to a point O^{\prime}(h, k) .
Let the new coordinates of P(x, y) be P\left(x^{\prime}, y^{\prime}\right) .
Then, x^{\prime}=x-h \Rightarrow x=x^{\prime}+h .
And, y^{\prime}=y-k \Rightarrow y=y^{\prime}+k .
So, the new equation becomes:
\begin{aligned}& \left(x^{\prime}+h\right)^{2}+\left(y^{\prime}+k\right)^{2}-4\left(x^{\prime}+h\right)-8\left(y^{\prime}+k\right)+3=0 \\ \\\Rightarrow & \left(x^{\prime 2}+h^{2}+2 x^{\prime} h\right)+\left(y^{\prime 2}+k^{2}+2 y^{\prime} k\right)-4\left(x^{\prime}+h\right)-8\left(y^{\prime}+k\right)+3=0\\ \\\Rightarrow & x^{\prime 2}+y^{\prime 2}+(2 h-4) x^{\prime}+(2 k-8) y^{\prime}+\left(h^{2}+k^{2}-4 h-8 k+3\right)=0 .\end{aligned}
Since we are required to get an equation free from first degree terms, so we have:
\begin{aligned}& (2 h-4=0 \text { and } 2 k-8=0)\\ \\\Rightarrow & (2 h=4 \text { and } 2 k=8) \Rightarrow (h=2 \text { and } k=4) .\end{aligned}Hence, the origin O should be shifted to the point O^{\prime}(2,4) .
Related Answered Questions
Question: 20.8.3
Verified Answer:
We have
\begin{aligned}\sqrt{3} x+y+2=0 \Ri...
Question: 20.12.6
Verified Answer:
The given lines are 2 x+3 y-4=0 a...
Question: 20.12.5
Verified Answer:
The equation of any line through the point of inte...
Question: 20.12.4
Verified Answer:
5 x-4 y+1=0 \Rightarrow y=\frac{5}{4} x...
Question: 20.12.3
Verified Answer:
5 x+4 y-20=0 \Rightarrow y=\frac{-5}{4} x+...
Question: 20.12.2
Verified Answer:
The given lines are 3 x+4 y-7=0 a...
Question: 20.12.1
Verified Answer:
The equation of any line through the point of inte...
Question: 20.11.4
Verified Answer:
Let the origin O be shifted to th...
Question: 20.11.3
Verified Answer:
Let O^{\prime}(h, k) be the point...
Question: 20.11.2
Verified Answer:
Let the original coordinates be P(x, y) [/...