P and Q are two points on the line through (2,4) and having slope m. If the line segment AB join

step 1 In this problem of conic section we have given that P and Q are two points on the line through 2

P and Q are two points on the line through (2,4) and having...

$\mathrm{P}$ and $\mathrm{Q}$ are two points on the line through $(2,4)$ and having slope $\mathrm{m}$. If the line segment $\mathrm{AB}$ joining $(0,0)$ and $(6,0)$ subtends right angles at $\mathrm{P}$ and $\mathrm{Q}$ then (a) $\mathrm{m} \notin\left[\frac{2-3 \sqrt{2}}{4}, \frac{2+3 \sqrt{2}}{4}\right]$ (b) $\frac{2-3 \sqrt{2}}{4}<\mathrm{m}<\frac{2+3 \sqrt{2}}{4}$ (c) $\frac{2-3 \sqrt{2}}{4} \leq \mathrm{m} \leq \frac{2+3 \sqrt{2}}{4}$ (d) $\mathrm{m} \in \mathrm{R}$

$\mathrm{P}$ and $\mathrm{Q}$ are two points on the line through $(2,4)$ and having slope $\mathrm{m}$. If the line segment $\mathrm{AB}$ joining $(0,0)$ and $(6,0)$ subtends right angles at $\mathrm{P}$ and $\mathrm{Q}$ then
(a) $\mathrm{m} \notin\left[\frac{2-3 \sqrt{2}}{4}, \frac{2+3 \sqrt{2}}{4}\right]$
(b) $\frac{2-3 \sqrt{2}}{4}<\mathrm{m}<\frac{2+3 \sqrt{2}}{4}$
(c) $\frac{2-3 \sqrt{2}}{4} \leq \mathrm{m} \leq \frac{2+3 \sqrt{2}}{4}$
(d) $\mathrm{m} \in \mathrm{R}$

Key Concepts

Discriminant of a Quadratic Equation

The discriminant of the quadratic equation obtained from the substitution indicates the nature of the intersections: if it is positive, there are two distinct intersection points; if zero, there is precisely one point (the line is tangent to the circle); and if negative, there is no real intersection. This factor is instrumental in establishing the range of values for parameters such as the slope in problems involving intersections.

Thales' Theorem

This theorem states that any angle inscribed in a semicircle (or subtended by a diameter of a circle) is a right angle. It is fundamental in connecting the geometry of circles with right angles, as it tells that if a segment is a diameter of a circle then the angle subtended at any point on the circle is a right angle.

Circle Defined by a Diameter

The circle defined by a given diameter consists of all points from which the endpoints of that diameter are seen under a right angle. This concept is central to many geometric problems that involve right angles and circles, as it provides a precise locus of points satisfying the angle condition.

Equation of a Line

The linear equation in point-slope form highlights how the slope and a specific point (or points) determine the unique line's orientation and position in the plane. Understanding the line's equation is crucial when analyzing its intersection with other curves such as circles.

Intersection of a Line and a Circle

Finding the intersection of a line and a circle involves substituting the line’s equation into the circle’s equation, which yields a quadratic equation in one variable. Analyzing the solutions of this quadratic gives the points of intersection, if any, and helps determine conditions on the line’s parameters for these intersections to exist.

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