If a, b, c are in A.P., prove that: (a-c)^2=4(a-b)(b-c) - Vidyadip

Question

If a, b, c are in A.P., prove that:
$(\text{a}-\text{c})^2=4(\text{a}-\text{b})(\text{b}-\text{c})$

✓

Answer

If $(\text{a}-\text{c})^2=4(\text{a}-\text{b})(\text{b}-\text{c})$
Then,
$\text{a}^2+\text{c}^2-2\text{ac}=4(\text{ab})-\text{b}^2-\text{ac}+\text{bc}$
$\Rightarrow\text{a}^2+\text{c}^24\text{b}^2+2\text{ac}-4\text{ac}-4\text{bc}=0$
$\Rightarrow(\text{a}+\text{c}-2\text{b})^2=0$ $\big[$ Using $(\text{a}+\text{b}+\text{c})^2=\text{a}^2+\text{b}^2+\text{c}^2+2\text{ab}+2\text{ac}+2\text{bc}\big]$
$\therefore\text{a}+\text{c}-2\text{b}=0$
or $\text{a}+\text{c}=2\text{b}$
and since,
$\text{a},\ \text{b},\ \text{c}$ are in A.P [Given]
$\text{a}+\text{c}=2\text{b}$
Hence proved
$(\text{a}-\text{b})^2=4(\text{a}-\text{b})(\text{b}-\text{c})$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Arithmetic Progressions→Arithmetic Progressions — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Determine whether the point (-3, 2) lies inside or outside the triangle whose sides are given by the equations x + y - 4 = 0, 3x - 7y + 8 = 0, 4x - y - 31 = 0.

Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:
$4\text{x}^2+3\text{y}^2=1$

Solve the equation $|\text{z}|=\text{z}+1+2\text{i}.$

Solve the following equations:
$2\sin^{2}\text{x}=3\cos\text{x},0\leq\text{x}\leq2\pi$

Solve the following system of equations in R.
$2\text{x}+6\geq0,4\text{x}-7<0$

Find the least positive integral value of n for which $\Big(\frac{1+\text{i}}{1-\text{i}}\Big)^{\text{n}}$ is real.

Find the mean deviation from the median for the following data:

Mark obtained	10	11	12	14	15
No. of students	2	3	8	3	4

Prove that: $\frac{\sin\text{(A-B)}}{\cos\text{A}\cos\text{B}}+\frac{\sin\text{(B-C)}}{\cos\text{B}\cos\text{C}}+\frac{\sin\text{(C-A)}}{\cos\text{C}\cos\text{A}}=0$

Find the value of (1.01)¹⁰ + (1 - 0.01)¹⁰ correct to 7 places of decimal.

A person observes the angle of elevation of the peak of a hill from a station to be $\alpha.$ He walks c metres along a slope inclined at an angle $\beta$ and finds the angle of elevation of the peak of the hill to be $\gamma.$ Show that the height of the peak above the ground is $\frac{\text{c}\sin\alpha\sin(\gamma-\beta)}{(\sin\gamma-\alpha)}.$