If m_i, 1/m_i, m_i > 0, i = 1, 2, 3, 4...

MathematicsStandard and General Equation of a CircleJEE Advanced 1989Moderate

If $m_{i}, \frac{1}{m _{i}}, m_{i} > 0, i = 1, 2, 3, 4$ are four distinct points on a circle, then show that $m_{1} m_{2} m_{3} m_{4} = 1$ .

Visualized Solution (English)

Visualizing the Problem

Consider four distinct points $(m_{i}, \frac{1}{m _{i}})$ for $i = 1, 2, 3, 4$ .
These points lie on the rectangular hyperbola $x y = 1$ .
They also lie on a circle whose equation we need to define.

The General Circle Equation

Let the general equation of the circle be:
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
where $g, f, c$ are constants.

Defining the Points

The points are of the form $(m, \frac{1}{m})$ .
Since these points lie on the circle, they must satisfy its equation.
We have four such values: $m_{1}, m_{2}, m_{3}, m_{4}$ .

Substitution Step

Substitute $x = m$ and $y = \frac{1}{m}$ into the circle equation:
$m^{2} + (\frac{1}{m})^{2} + 2 g (m) + 2 f (\frac{1}{m}) + c = 0$
This simplifies to: $m^{2} + \frac{1}{m ^{2}} + 2 g m + \frac{2 f}{m} + c = 0$

Clearing the Denominators

To eliminate the fractions, multiply the entire equation by $m^{2}$ :
$m^{2} (m^{2} + \frac{1}{m ^{2}} + 2 g m + \frac{2 f}{m} + c) = 0$
Expanding this gives: $m^{4} + 1 + 2 g m^{3} + 2 f m + c m^{2} = 0$

The Quartic Equation Form

Rearrange the terms in descending order of $m$ :
$m^{4} + 2 g m^{3} + c m^{2} + 2 f m + 1 = 0$
This is a quartic equation in $m$ .

Applying Vieta's Formulas

For a quartic equation $a x^{4} + b x^{3} + c x^{2} + d x + e = 0$ , the product of roots is given by $\frac{e}{a}$ .
In our equation: $a = 1$ and $e = 1$ .
Therefore, the product of roots $m_{1} m_{2} m_{3} m_{4} = \frac{1}{1}$ .

Final Result

The product of the four distinct values is:
$m_{1} m_{2} m_{3} m_{4} = 1$
Hence proved.

The Way Forward

Key Takeaway: The intersection of a circle and a rectangular hyperbola leads to a quartic equation where the product of the x-coordinates is related to the constant term.
Next Challenge: What if the points were on a parabola $y^{2} = 4 a x$ ? How would the product of the ordinates behave?

00:00 / 00:00

Conceptually Similar Problems

MathematicsStandard and General Equation of a CircleJEE Advanced 1997Moderate

View in:English Hindi

Let $C$ be any circle with centre $(0, 2)$ . Prove that at the most two rational points can be there on $C$ . (A rational point is a point both of whose coordinates are rational numbers.)

MathematicsRectangular HyperbolaJEE Advanced 1998Moderate

View in:English Hindi

If the circle $x^{2} + y^{2} = a^{2}$ intersects the hyperbola $x y = c^{2}$ in four points $P (x_{1}, y_{1}), Q (x_{2}, y_{2}), R (x_{3}, y_{3}), S (x_{4}, y_{4})$ , then

* Multiple Correct Options

(A)

x_{1} + x_{2} + x_{3} + x_{4} = 0

(B)

y_{1} + y_{2} + y_{3} + y_{4} = 0

(C)

x_{1} x_{2} x_{3} x_{4} = c^{4}

(D)

y_{1} y_{2} y_{3} y_{4} = c^{4}

Answer:A, B, C, D

MathematicsEquation of Tangent and NormalJEE Advanced 1998Moderate

View in:English Hindi

$C_{1}$ and $C_{2}$ are two concentric circles, the radius of $C_{2}$ being twice that of $C_{1}$ . From a point $P$ on $C_{2}$ , tangents $P A$ and $P B$ are drawn to $C_{1}$ . Prove that the centroid of the triangle $P A B$ lies on $C_{1}$ .

MathematicsVector (Cross) ProductJEE Advanced 1987Moderate

View in:English Hindi

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

MathematicsProperties of TrianglesJEE Advanced 2000Difficult

View in:English Hindi

Let $A B C$ be a triangle with incentre $I$ and inradius $r$ . Let $D, E, F$ be the feet of the perpendiculars from $I$ to the sides $B C, C A$ and $A B$ respectively. If $r_{1}, r_{2}$ and $r_{3}$ are the radii of circles inscribed in the quadrilaterals $A F I E, B D I F$ and $C E I D$ respectively, prove that $\frac{r _{1}}{r - r _{1}} + \frac{r _{2}}{r - r _{2}} + \frac{r _{3}}{r - r _{3}} = \frac{r _{1} r _{2} r _{3}}{( r - r _{1} ) ( r - r _{2} ) ( r - r _{3} )}$ .

MathematicsProperties of TrianglesJEE Advanced 1996Moderate

View in:English Hindi

A circle passes through three points $A, B$ and $C$ with the line segment $A C$ as its diameter. A line passing through $A$ intersects the chord $B C$ at a point $D$ inside the circle. If angles $D A B$ and $C A B$ are $α$ and $β$ respectively and the distance between the point $A$ and the mid point of the line segment $D C$ is $d$ , prove that the area of the circle is $\frac{π d ^{2} c o s ^{2} α}{c o s ^{2} α + c o s ^{2} β + 2 c o s α c o s β c o s ( β - α )}$ .

MathematicsEquation of Tangent and NormalJEE Advanced 2005Moderate

View in:English Hindi

Circles with radii 3, 4 and 5 touch each other externally. If $P$ is the point of intersection of tangents to these circles at their points of contact, find the distance of $P$ from the points of contact.

Answer:2.236

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1981Moderate

View in:English Hindi

Let the complex number $z_{1}, z_{2}, z_{3}$ be the vertices of an equilateral triangle. Let $z_{0}$ be the circumcentre of the triangle. Then prove that $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3 z_{0}^{2}$ .

MathematicsCube Roots and nth Roots of UnityJEE Advanced 1984Moderate

View in:English Hindi

If $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are the $n$ roots of unity, then show that $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1}) = n$ .

MathematicsDistance and Section FormulasJEE Advanced 1997Moderate

View in:English Hindi

Let $S$ be a square of unit area. Consider any quadrilateral which has one vertex on each side of $S$ . If $a, b, c$ , and $d$ denote the lengths of the sides of the quadrilateral, prove that $2 \leq a^{2} + b^{2} + c^{2} + d^{2} \leq 4$ .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If mi​,mi​1​,mi​>0,i=1,2,3,4 are four distinct points on a circle, then show that m1​m2​m3​m4​=1.

Visualized Solution (English)

Visualizing the Problem

The General Circle Equation

Defining the Points

Substitution Step

Clearing the Denominators

The Quartic Equation Form

Applying Vieta's Formulas

Final Result

The Way Forward

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the circle x2+y2=a2 intersects the hyperbola xy=c2 in four points P(x1​,y1​),Q(x2​,y2​),R(x3​,y3​),S(x4​,y4​), then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B,C,D are any four points in space, prove that ∣AB×CD+BC×AD+CA×BD∣=4(area of triangle ABC).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 1,a1​,a2​,…,an−1​ are the n roots of unity, then show that (1−a1​)(1−a2​)(1−a3​)…(1−an−1​)=n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If mi​,mi​1​,mi​>0,i=1,2,3,4 are four distinct points on a circle, then show that m1​m2​m3​m4​=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the circle x2+y2=a2 intersects the hyperbola xy=c2 in four points P(x1​,y1​),Q(x2​,y2​),R(x3​,y3​),S(x4​,y4​), then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B,C,D are any four points in space, prove that ∣AB×CD+BC×AD+CA×BD∣=4(area of triangle ABC).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If 1,a1​,a2​,…,an−1​ are the n roots of unity, then show that (1−a1​)(1−a2​)(1−a3​)…(1−an−1​)=n.

If $m_{i}, \frac{1}{m _{i}}, m_{i} > 0, i = 1, 2, 3, 4$ are four distinct points on a circle, then show that $m_{1} m_{2} m_{3} m_{4} = 1$ .

If the circle $x^{2} + y^{2} = a^{2}$ intersects the hyperbola $x y = c^{2}$ in four points $P (x_{1}, y_{1}), Q (x_{2}, y_{2}), R (x_{3}, y_{3}), S (x_{4}, y_{4})$ , then

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

If $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are the $n$ roots of unity, then show that $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1}) = n$ .

If $m_{i}, \frac{1}{m _{i}}, m_{i} > 0, i = 1, 2, 3, 4$ are four distinct points on a circle, then show that $m_{1} m_{2} m_{3} m_{4} = 1$ .

If the circle $x^{2} + y^{2} = a^{2}$ intersects the hyperbola $x y = c^{2}$ in four points $P (x_{1}, y_{1}), Q (x_{2}, y_{2}), R (x_{3}, y_{3}), S (x_{4}, y_{4})$ , then

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

If $1, a_{1}, a_{2}, \dots, a_{n - 1}$ are the $n$ roots of unity, then show that $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1}) = n$ .