Explanation

Given hyperbola: y² = (x/a)² + 1 → y² - x²/a² = 1

This is of form y²/a₀² - x²/b² = 1 with transverse axis along y-axis.

Here a₀² = 1, b² = a²

For hyperbola: c² = a₀² + b², foci at (0, ±c)

Given one focus = (0, √2) → c = √2 → c² = 2

So 2 = 1 + a²  

→ a² = 1  

→ a = ±1

Usually we take a > 0, so a = 1

Last verified on 05-06-2026

Explanation

Last verified on 04-06-2026

Explanation

1. AB is one side → D = A + C - B = (-1+1-6, 2-20-2) = (-6, -20)
2. AC is one side → D = A + B - C = (-1+6-1, 2+2+20) = (4, 24)
3. BC is one side → D = B + C - A = (6+1+1, 2-20-2) = (8, -20)

Explanation

(1 + i)¹⁰ = (√2)¹⁰ × [cos(45°×10) + i sin(45°×10)]

           = 2⁵ × [cos450° + i sin450°]

           = 32 × [cos90° + i sin90°] 450° = 360° + 90°

           = 32 × [0 + i×1]

           = 32i

Last verified on 04-06-2026

Explanation

The Fourier series expansion of an even, periodic function contains only cosine terms and a constant term.

In other words, the sine terms are absent.

Even functions:

An even function is symmetric about the y-axis, meaning f(x) = f(-x). 

Fourier series:

A Fourier series represents a periodic function as a sum of sine and cosine waves. 

Explanation

Explanation

y = (1/8)x^2 + (1/2)x + 1

At (0,1)

1) Slope of tangent at (0,1) 

y = (1/8)x² + (1/2)x + 1  

y' = (1/4)x + 1/2  

At x=0, m = 1/2

2) Tangent equation

y - 1 = (1/2)x → y = 0.5x + 1

3) Intercepts

x-intercept: 0 = 0.5x + 1 → x = -2  

y-intercept: y = 1  

4) Area 

Triangle with base 2, height 1  

Area = 1/2 × 2 × 1 = 1

Last verified on 04-06-2026

Explanation

Explanation

A triangle is classified based on its angles.

If one angle is 90°, it is a right-angled triangle.

The given triangle has angles 30°, 60°, and 90°.

Since it contains a 90° angle, it is a right triangle.

Last verified on 25-05-2026

Explanation

A secant is a straight line that intersects a circle.

It passes through the circle and cuts it at two points.

Last verified on 25-05-2026