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If h denotes the arithmetic mean and k denotes the geometric mean of the intercepts made on the coordinate axes by the line passing through the point$\left( {1,1} \right)$, then point $\left( {h,k} \right)$ lies on
A) circle
B) A parabola
C) A straight line
D) A hyperbola

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Last updated date: 13th Jun 2024
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Answer
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Hint: The equation of the straight line in slope-intercept is used in this question.
Equation of line in intercept form is given by $\dfrac{x}{a} + \dfrac{y}{b} = 1$ , where a and b are intercepts made on the x and y axis respectively.

Complete step by step answer:
Let’s assume the equation of straight line in intercept form is given by,
$\dfrac{x}{a} + \dfrac{y}{b} = 1......(1)$

The line passes through the point $\left( {1,1} \right)$.It means it satisfies the equation (1). Substituting the value $x = 1$ and $y = 1$ in equation (1),
$
  \dfrac{1}{a} + \dfrac{1}{b} = 1 \\
  \dfrac{{a + b}}{{ab}} = 1......(2) \\
 $

Now, the arithmetic mean between the two numbers c and d is given by,
$AM = \dfrac{{c + d}}{2}$

According to the question, h is the arithmetic mean between the intercepts a and b made on the x and y-axis respectively.
$
  h = \dfrac{{a + b}}{2} \\
  a + b = 2h......(3) \\
 $

Now, the geometric mean between the two numbers c and d is given by,
$GM = \sqrt {cd} $

According to the question, k is the geometric mean between the intercepts a and b made on the x and y-axis respectively.
$k = \sqrt {ab} ......(4)$

Squaring both sides of equation (4),
${k^2} = ab......(5)$

Now, substitute the value of (a+b) and (ab) in equation (2),
$
  \dfrac{{2h}}{{{k^2}}} = 1 \\
  {k^2} = 2h......(6) \\
 $
lies on the curve. It means it should satisfy the equation of the curve.
If and , substitute it in equation (6)
${y^2} = 2x$

This is an equation of a parabola.
Hence, the correct option is (B).

Note:
 There are 3 important steps in this question: (1) equation of a line in intercept form, (2) arithmetic mean and geometric mean between two numbers, and (3) knowledge of the standard equations of various conic sections.
Standard equation of various conic sections
For parabola the standard equation is ${y^2} = 4ax$
For hyperbola the standard equation is $\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1$
For ellipse the standard equation is $\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$
For circle the standard equation is ${x^2} + {y^2} = {r^2}$