Let the maximum and minimum values of ( 8 x – x 2 – 12 – 4 ) 2 + ( x - 7 ) 2 , x ? R be M and m, respectively. Then M 2 – m 2 is equal to __________. [2024]

Question

Let the maximum and minimum values of ${(\sqrt{8 x - x^{2} - 12} - 4)}^{2} + {(x - 7)}^{2}, x \in R$ be M and m, respectively. Then $M^{2} - m^{2}$ is equal to __________. [2024]

Smart Achievers · Accepted Answer

(1600)

Let $y = \sqrt{8 x - x^{2} - 12}$

$\Rightarrow y^{2} = 8 x - x^{2} - 12$

$= - (x^{2} - 8 x + 16) + 4 = - {(x - 4)}^{2} + 4$

$\Rightarrow {(x - 4)}^{2} + y^{2} = 4$ ... (i)

which is a circle with centre (4, 0) and radius 2.

Now, ${(y - 4)}^{2} + {(x - 7)}^{2}$ represent distance of (x, y) from (7, 4)

M = Maxium distance = ${(2 - 7)}^{2} + {(0 - 4)}^{2} = 25 + 16 = 41$

m = Minimum distance = Distance between P and (7, 4)

where P is the intersection of circle with line joining (4, 0) and (7, 4).

Now, equation of line joining (4, 0) and (7, 4) is given by

$(y - 0) = \frac{4 - 0}{7 - 4} (x - 4)$

i.e., $y = \frac{4}{3} (x - 4)$

On substituting the value of y is in (i), we get

${(x - 4)}^{2} + \frac{16}{9} {(x - 4)}^{2} = 4$

$\Rightarrow {(x - 4)}^{2} [\frac{25}{9}] = 4 \Rightarrow {(x - 4)}^{2} = \frac{36}{25} \Rightarrow x - 4 = \pm \frac{6}{5}$

$\Rightarrow x = \frac{6}{5} + 4 = \frac{26}{5}, y = \frac{8}{5}$ (Neglect negative sign)

So, $m = {(\frac{26}{5} - 7)}^{2} + {(\frac{8}{5} - 4)}^{2} = \frac{81}{25} + \frac{144}{25} = \frac{225}{25} = 9$

$∴ M^{2} - m^{2} = 41^{2} - 9^{2} = 1600$ .

Solution

Q.
Let the maximum and minimum values of ${(\sqrt{8 x - x^{2} - 12} - 4)}^{2} + {(x - 7)}^{2}, x \in R$ be M and m, respectively. Then $M^{2} - m^{2}$ is equal to __________. [2024]

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let the maximum and minimum values of (8x–x2–12–4)2+(x-7)2, x∈R be M and m, respectively. Then M2–m2 is equal to __________. [2024]

Want Us to Email you About Special Offers & Updates?

Q.
Let the maximum and minimum values of ${(\sqrt{8 x - x^{2} - 12} - 4)}^{2} + {(x - 7)}^{2}, x \in R$ be M and m, respectively. Then $M^{2} - m^{2}$ is equal to __________. [2024]