Circle Intersection Calculation
We are given two circles and need to find a point of intersection.
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Define Circle Equations:
- Circle 1: Center $(0.5, 0)$, radius $0.5$. Equation: $\\(x - 0.5)^2 + y^2 = (0.5)^2 \\)$
- Circle 2: Center $(1, 1)$, radius $1$. Equation: $\\(x - 1)^2 + (y - 1)^2 = 1^2 \\)$
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Simplify Equations:
- Circle 1: $\\(x^2 - x + 0.25 + y^2 = 0.25 \\) \implies x^2 - x + y^2 = 0 \\) (1)$
- Circle 2: $\\(x^2 - 2x + 1 + y^2 - 2y + 1 = 1 \\) \implies x^2 - 2x + y^2 - 2y + 1 = 0 \\) (2)$
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Find the Radical Axis Equation: Subtract equation (1) from equation (2).
$\\( (x^2 - 2x + y^2 - 2y + 1) - (x^2 - x + y^2) = 0 \\)$
$\\{-x - 2y + 1 = 0 \\)$
Solve for $x$: $\\(x = 1 - 2y \\)$
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Substitute into Circle Equation (1): Replace $x$ in $\\(x^2 - x + y^2 = 0 \\)$.
$\\( (1 - 2y)^2 - (1 - 2y) + y^2 = 0 \\)$
$\\( (1 - 4y + 4y^2) - 1 + 2y + y^2 = 0 \\)$
$\\(5y^2 - 2y = 0 \\)$
Factor: $\\(y(5y - 2) = 0 \\)$
Possible $y$ values: $\\(y = 0 \\)$ or $\\(y = 2/5 = 0.4 \\)$.
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Calculate Corresponding $x$ Values: Use $\\(x = 1 - 2y \\)$.
- If $\\(y = 0 \\)$, then $\\(x = 1 - 2(0) = 1 \\)$. Point: $(1, 0)$.
- If $\\(y = 0.4 \\)$, then $\\(x = 1 - 2(0.4) = 1 - 0.8 = 0.2 \\)$. Point: $(0.2, 0.4)$.
- Identify Intersection Point: The points of intersection are $(1, 0)$ and $(0.2, 0.4)$. The point $\\( (0.2, 0.4) \\)$ matches option B.
Important Questions from Mensuration and Geometry
In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm²) of the rectangle PLMN?
Note: The figure shown is representative.
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $r$ cm as shown in the figure. The side of the dodecagon is $d$ cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side $r$ cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
- Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?
- During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be $89.85^{\circ}$, the ratio of the Earth-Sun and Earth-Moon distances is closest to
Let $S$ be the portion of the plane $z = 2x + 2y - 100$ which lies inside the cylinder $x^2 + y^2 = 1$. If the surface area of $S$ is $\alpha\pi$, then the value of $\alpha$ is equal to ________.
