If |z+1|+|z−3|≤10, then the range of values of |z−7| is

|z+1|+|z−3|≤10 the ellipse with foci at (-1,0), and (3,0) and length of major axis 10. Its centre is (1,0), and its equation is (x−1)225+y221=1Any point on the ellipse is P(1+5cosθ,21sinθ).Its distance from A(7,0) is given byAP2=(5 cos θ−6)2+21 sin2θ=25cos2θ−60cosθ+36+21sin2θ=21+36+4cos2θ−60cosθ=(2 cos θ−15)2−152+57⇒132−152+57≤AP2≤172−152+57⇒226−225≤AP2≤64+57⇒1≤AP2≤121⇒1≤AP≤11

If |z+1|+|z−3|≤10, then the range of values of |z−7| is

|z+1|+|z−3|≤10 the ellipse with foci at (-1,0), and (3,0) and length of major axis 10. Its centre is (1,0), and its equation is (x−1)225+y221=1Any point on the ellipse is P(1+5cosθ,21sinθ).Its distance from A(7,0) is given byAP2=(5 cos θ−6)2+21 sin2θ=25cos2θ−60cosθ+36+21sin2θ=21+36+4cos2θ−60cosθ=(2 cos θ−15)2−152+57⇒132−152+57≤AP2≤172−152+57⇒226−225≤AP2≤64+57⇒1≤AP2≤121⇒1≤AP≤11

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If $| z + 1 | + | z - 3 | \leq 10$ , then the range of values of $| z - 7 |$ is

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$[1, 11]$

$[2, 12]$

$[7, 9]$

$[0, 10]$

answer is B.

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Detailed Solution

$| z + 1 | + | z - 3 | \leq 10$ the ellipse with foci at (-1,0), and (3,0) and length of major axis 10. Its centre is (1,0), and its equation is $\frac{{(x - 1)}^{2}}{25} + \frac{y^{2}}{21} = 1$

Any point on the ellipse is $P (1 + 5 \cos θ, \sqrt{21} \sin θ) .$

Its distance from A(7,0) is given by

$\begin{array}{l} A P^{2} = {(5 \cos θ - 6)}^{2} + 21 \sin^{2} θ \\ = 25 \cos^{2} θ - 60 \cos θ + 36 + 21 \sin^{2} θ \\ = 21 + 36 + 4 \cos^{2} θ - 60 \cos θ \\ = {(2 \cos θ - 15)}^{2} - 15^{2} + 57 \\ \Rightarrow 13^{2} - 15^{2} + 57 \leq A P^{2} \leq 17^{2} - 15^{2} + 57 \\ \Rightarrow 226 - 225 \leq A P^{2} \leq 64 + 57 \\ \Rightarrow 1 \leq A P^{2} \leq 121 \\ \Rightarrow 1 \leq A P \leq 11 \end{array}$