If $a^{x_{1}} = b^{x_{2}} = c^{x_{3}}$ where $x_{1}, x_{2}, x_{3}$ are unequal positive numbers and a, b, c are in G.P. then $x_{1}, x_{2}, x_{3}$ are in HP

True

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False

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Solution

The correct option is A True
$l e t a = a, b = a r a n d c = a r^{2} t h e n a^{x_{1}} = (a r)^{x_{2}} = (a r^{2})^{x_{3}} = k t h e n x_{1} = (\frac{k}{l o g a}) x_{2} = (\frac{k}{l o g (a r)}) x_{3} = (\frac{k}{l o g (a r^{2})}) ∴ (\frac{1}{x_{1}}) = (\frac{l o g a}{k}), (\frac{1}{x_{2}}) = (\frac{l o a + l o g r}{k}) a n d (\frac{1}{x_{3}}) = (\frac{l o g a + 2 l o g r}{k}) c l e a r l y (\frac{1}{x_{1}}), (\frac{1}{x_{2}}), (\frac{1}{x_{3}}) a r e i n A P ∴ x_{1}, x_{2}, x_{3} a r e i n H P$

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Similar questions

Q. If

a^{x_{1}} = b^{x_{2}} = c^{x_{3}}

where

x_{1}, x_{2}, x_{3}

are unequal positive numbers and a, b, c are in G.P., then

x_{1}^{3} + x_{3}^{3}

Q. If

A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3})

are the vertices of an equilateral triangle and

(x_{1} - 4)^{2} + (y_{1} - 5)^{2} = (x_{2} - 4)^{2} + (y_{2} - 5)^{2} = (x_{3} - 4)^{2} + (y_{3} - 5)^{2},

then the value of

(y_{1} + y_{2} + y_{3}) - (x_{1} + x_{2} + x_{3})

Find the centroid of the triangle with vertices A $(x_{1}, y_{1})$ , B $(x_{2}, y_{2})$ and C $(x_{3}, y_{3})$ .

The vertices of a square ABCD are given as A (x $_{1}$ , y $_{1}$ ), B (x $_{2}$ , y $_{2}$ ), C (x $_{3}$ , y $_{3}$ ) and D (x $_{4}$ , y $_{4}$ ).
Which of the following will always hold true?

Q. Assertion :Three points

A (x_{1}, y_{1})

B (x_{2}, y_{2})

and

C (x_{3}, y_{3})

are collinear if

x_{1} + x_{2} + x_{3} = y_{1} + y_{2} + y_{3} .

Reason: The points

A (x, - x)

B (7, - 5)

C (- 5, 3)

are collinear if

x = 1

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If ax1=bx2=cx3 where x1,x2,x3 are unequal positive numbers and a, b, c are in G.P. then x1, x2, x3 are in HP

The correct option is A Trueleta=a,b=arandc=ar2thenax1=(ar)x2=(ar2)x3=kthenx1=(kloga)x2=(klog(ar))x3=(klog(ar2))∴(1x1)=(logak),(1x2)=(loa+logrk)and(1x3)=(loga+2logrk)clearly(1x1),(1x2),(1x3)areinAP∴x1,x2,x3areinHP

If $a^{x_{1}} = b^{x_{2}} = c^{x_{3}}$ where $x_{1}, x_{2}, x_{3}$ are unequal positive numbers and a, b, c are in G.P. then $x_{1}, x_{2}, x_{3}$ are in HP