If the lines joining the origin to the intersection of the line $ y = mx + 2 $ and the circle $ x^2 + y^2 = 1 $ are at right angles, then

Given equation of line $y = mx + 2$ or $\frac{y - mx}{2} = 1$ and circle $ x^2 + y^2 = 1$ $\therefore $ Equation of the line joining the origin to the intersection of line and circle is $ x^2 + y^2 - \left(\frac{y - mx}{2}\right)^{2} = 0 $ $ \Rightarrow 4\left(x^{2} + y^{2}\right) -\left(y^{2} + m^{2} x^{2} - 2myx\right) = 0$ $ \Rightarrow x^{2}\left(4-m^{2}\right) + 3y^{2 } + 2mxy= 0$ Since, lines are at right angles. $ \therefore 4 -m^{2 } + 3 = 0 \left( \because a + b = 0\right) $ $ \Rightarrow m^{2 } =7 $ $ \Rightarrow m = \pm\sqrt{7}$

If the lines joining the origin to the intersectio

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Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points.

A straight line can be represented as an equation in various forms, as show in the image below:

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y₀ = m (x – x₀)

2. Two – Point Form

Let's look at the line. L crosses between two places. P₁(x₁, y₁) and P₂(x₂, y₂) are general points on L, while P (x, y) is a general point on L. As a result, the three points P₁, P₂, and P are collinear, and it becomes

The slope of P₂P = The slope of P₁P₂, i.e.

$\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}$

Hence, the equation becomes:

y - y₁ =$ \frac{y_2-y_1}{x_2-x_1} (x-x1)$

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c