Match the elements of Column I to elements of Column II....

Match the elements of Column I to elements of Column II. There can be single or multiple matches. The line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent to Column I (a) the parabola $\mathrm{y}^{2}=4 \mathrm{x}$, if (b) the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$, if (c) the hyperbola $\frac{x^{2}}{12}-\frac{y^{2}}{9}=1$, if (d) the circle $x^{2}+y^{2}=3$, if Column II (p) $\mathrm{m}=2, \mathrm{c}=\pm \sqrt{15}$ (q) $\mathrm{m}=\frac{1}{\sqrt{5}}, \mathrm{c}=\pm 3$ (r) $\mathrm{m}=2, \mathrm{c}=\pm \sqrt{39}$ (s) $\quad \mathrm{m}=1, \mathrm{c}=1$

Match the elements of Column I to elements of Column II. There can be single or multiple matches.
The line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent to
Column I
(a) the parabola $\mathrm{y}^{2}=4 \mathrm{x}$, if
(b) the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$, if
(c) the hyperbola $\frac{x^{2}}{12}-\frac{y^{2}}{9}=1$, if
(d) the circle $x^{2}+y^{2}=3$, if
Column II
(p) $\mathrm{m}=2, \mathrm{c}=\pm \sqrt{15}$
(q) $\mathrm{m}=\frac{1}{\sqrt{5}}, \mathrm{c}=\pm 3$
(r) $\mathrm{m}=2, \mathrm{c}=\pm \sqrt{39}$
(s) $\quad \mathrm{m}=1, \mathrm{c}=1$

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The main types include the parabola, ellipse, hyperbola, and circle. Each conic has a standard equation with specific geometric properties that determine its shape and the criteria for tangency when intersected by a line.

Tangency Condition

A tangent to a curve is a line that touches the curve at exactly one point. In analytic geometry, this is typically achieved by substituting the line’s equation into the curve's equation to form a quadratic equation in one variable. For the line to be tangent, this quadratic must have exactly one solution, which is ensured by setting its discriminant to zero.

Quadratic Equation Discriminant

The discriminant of a quadratic equation in the form ax² + bx + c = 0, given by b² - 4ac, indicates the nature of the roots. When the discriminant is zero, the equation has a single repeated real root, which geometrically corresponds to a tangent line touching a curve at exactly one point.

Slope-Intercept Form of a Line

The equation of a line in slope-intercept form, y = mx + c, expresses the line in terms of its slope (m) and y-intercept (c). This form is particularly useful in problems involving tangency because the parameters m and c can be substituted into the equation of a conic to determine the conditions under which the line is tangent to the curve.

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