IIT-JEE MAINS 2027 cheat sheet for Coordinate Geometry! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Coordinate Geometry (IIT-JEE Mains 2027)

Coordinate Geometry connects algebra with geometric representation. Mastering this chapter requires a strong command over fundamental distance metrics, locus equations, and canonical forms of conics.

1. Straight Lines & Point Systems

The position of a point dividing a line segment joining \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) in the ratio \(m:n\) internally is given by the section formula:

\[\Bigl(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\Bigr)\]

The general equation of a straight line is \(Ax + By + C = 0\). The perpendicular distance from a point \((x_0, y_0)\) to this line is:

\[d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}\]

2. Circles

The standard equation of a circle centered at \((h, k)\) with radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). The general form is \(x^2 + y^2 + 2gx + 2fy + c = 0\), where the center is \((-g, -f)\) and radius is \(r = \sqrt{g^2 + f^2 - c}\).

The condition for tangency of the line \(y = mx + c\) to the circle \(x^2 + y^2 = a^2\) is given by:

\[c^2 = a^2(1 + m^2)\]

3. Parabola, Ellipse & Hyperbola

• Parabola: For \(y^2 = 4ax\), the focal distance of a point \(P(x, y)\) is \(x + a\). Tangents in slope form are \(y = mx + \frac{a}{m}\).
• Ellipse: For \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (where \(a > b\)), eccentricity is \(e = \sqrt{1 - \frac{b^2}{a^2}}\).
• Hyperbola: For \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), eccentricity is \(e = \sqrt{1 + \frac{b^2}{a^2}}\).