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Chapter 14: Problem 1223

If \(\mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)\) and \(\mathrm{P}\left(\mathrm{tx}_{2}+(1-\mathrm{t}) \mathrm{x}_{1}, \mathrm{t}_{2}+(1-\mathrm{t}) \mathrm{y}_{1}\right)\) where \(t<0\), then P divides \(\underline{A B}\) from \(A\) in the ratio \(\ldots \ldots\) (a) \(1-\mathrm{t}\) (b) \([(\mathrm{t}-1) / \mathrm{t}]\) (c) \([t /(1-t)]\) (d) \(t-1\)

Short Answer

Expert verified
None of the given options match the correct ratio: \(\frac{1}{t}\).

Step by step solution

01

Write down the given coordinates

Point A has coordinates \(A(x_1, y_1)\), point B has coordinates \(B(x_2, y_2)\), and point P has coordinates \(P(tx_2+(1-t)x_1, ty_2+(1-t)y_1)\).
02

Apply section formula

The section formula states that if a point divides a line segment in the ratio \(m : n\), then its coordinates are given by the formula: \[\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\] Here, we need to find the ratio m:n in which P divides AB from A.
03

Compare the coordinates of P with the section formula

To find the ratio in which P divides AB, we need to compare the P's coordinates with the section formula: \(P(tx_2+(1-t)x_1, ty_2+(1-t)y_1)\). Comparing this with the section formula \(\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}\right)\), we can write: \[m+n=1 \quad \text{and} \quad \frac{m}{n}=t\]
04

Solve the equations for the ratio

Let's solve the equations for the ratio: \[ \frac{m+n}{m}=t\] Now substitute the value of \(m+n=1\), we get: \[\frac{1}{m}=t\] Now, solve for the ratio: \[m = \frac{1}{t}\]
05

Find the correct answer choice

Since we found that \(m = \frac{1}{t}\), we can now determine the correct answer choice: (a) \(1-t\) - This option is incorrect as the ratio is not in the form of \(1 - t\). (b) \[\frac{(t-1)}{t}\] - This option is almost correct, but the numerator should be \(1\), not \((t-1)\). (c) \[\frac{t}{(1-t)}\] - This option is incorrect as the numerator should be \(1\), not \(t\). (d) \(t-1\) - This option is incorrect as the ratio is not in the form of \(t - 1\). However, none of the given options match our result. We got the ratio to be \(m = \frac{1}{t}\), but this option is not provided. So there might be an error in the problem statement or the answer choices.

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Most popular questions from this chapter

The equation of line passing through the point \((-5,4)\) and making the intercept of length \((2 / \sqrt{5})\) between the lines \(x+2 y-1=0\) and \(x+2 y+1=0\) is \(\ldots \ldots\) (a) \(2 \mathrm{x}-\mathrm{y}+4=0\) (b) \(2 x-y-14=0\) (c) \(2 \mathrm{x}-\mathrm{y}+14=0\) (d) None of these

If the lines \(\mathrm{x}+\mathrm{y}+\mathrm{r}=0\) and \(\lambda \mathrm{x}-5 \mathrm{y}=5\) are identical then \(\lambda+\mathrm{r}=\ldots \ldots\) (a) \(-4\) (b) 4 (c) 1 (d) \(-1\)

If the point \([1+(t / \sqrt{2}), 2+(t / \sqrt{2})]\) lies between the two parallel lines \(\mathrm{x}+2 \mathrm{y}=1\) and \(2 \mathrm{x}+4 \mathrm{y}=15\), then the range of \(\mathrm{t}\) is \(\ldots \ldots\) (a) \(0<\mathrm{t}<[5 /(6 \sqrt{2})]\) (b) \(-[(4 \sqrt{2}) / 3]<\mathrm{t}<0\) (c) \(-[(4 \sqrt{2}) / 3]<\mathrm{t}<[(5 \sqrt{2}) / 6]\) (d) None of these

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The line parallel to the \(X\) -axis and passing through the intersection of the lines \(\mathrm{ax}+2 \mathrm{by}+3 \mathrm{~b}=0\) and \(\mathrm{bx}-2 \mathrm{ay}-3 \mathrm{a}=0\) where \((\mathrm{a}, \mathrm{b}) \neq(0,0)\) is: (A) above the X-axis at a distance of \((2 / 3)\) from it (B) above the X-axis at a distance of \((3 / 2)\) from it (C) below the X-axis at a distance of \((2 / 3)\) from it (D) below the X-axis at a distance of \((3 / 2)\) from it
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