Question

Given the quadratic expression x²+5x+4, which of the following are factors? Choose all answers that apply:

A) x+1
B) x+4
C) 2x+5
D) x²+5x

Answer 1

The locus of the point of intersection of tangents drawn to the circles x² + y² = a² and x² + y² = b², which are perpendicular to each other, is a circle with its center at the origin and a radius of √(a² * b²).

Explanation:

To prove this, let's consider the equations of the circles x² + y² = a² and x² + y² = b². For any point (x₀, y₀) on these circles, the equation of the tangent at that point can be written as xx₀ + yy₀ = a² and xx₀ + yy₀ = b² respectively, where (x, y) is the point of tangency.

Given that the tangents drawn to these circles are perpendicular, the product of the slopes of these tangents is -1. Hence, the product of the slopes of the lines xx₀ + yy₀ = a² and xx₀ + yy₀ = b² is -1.

Considering the equations of the tangents, the product of their slopes is (1/x₀² + 1/y₀²) = -1. Solving this yields x₀² * y₀² = -1. Now, for a point (x₀, y₀) to lie on the locus of the intersection of these tangents, x₀² * y₀² must be a constant, say k.

From the above relation, x₀² * y₀² = k = a² * b². Therefore, the locus of the point of intersection of the perpendicular tangents is a circle centered at the origin (0,0) with a radius √(a² * b²). This circle is concentric with the given circles x² + y² = a² and x² + y² = b².

Answer 2

The correct factor of the quadratic expression x²+5x+4 from the given options is B) x+4. This can be determined by factoring the quadratic expression into two binomials that multiply to give the original expression.

Step-by-step explanation:

The quadratic expression x²+5x+4 can be factored to determine which of the given options are the correct factors of the expression. Factoring a quadratic expression involves finding two binomials that when multiplied together will give the original quadratic expression. Factors of the constant term, 4, need to be considered in pairs such that their product is 4 and their sum is the coefficient of the middle term, 5.

Considering the options given:

1. x+1: When we multiply (x+1) with another term, the constant has to be 4 in order to get a product of 4. However, adding this 4 to the x term from (x+1) would not give us the middle term of 5x.
2. x+4: This is a factor, as (x+1)(x+4) gives us the original expression x²+5x+4.
3. 2x+5: This cannot be a factor as none of the factors of 4 can produce a term of 2x when multiplied by x.
4. x²+5x: This is just a part of the original expression and not a factor in the sense of being a binomial that multiplies with another to give the quadratic expression.

Thus, the correct factor from the options provided is B) x+4.